LIBRARY OF CONGRESS, 
Shelf 3*1. 



UNITED STATES OF AMERICA. 



PLANE AND SPHERICAL 



TRIGONOMETRY. 



A BY 

EDWARD BROOKS, A.M., Ph.D., 

Superintendent of Public Instruction in Philadelphia; 

Author of "Normal Series of Mathematics," "Normal Methods 

of Teaching," "Mental Science and Culture," 

"Philosophy of Arithmetic," etc. 






7 



^v $ 



PHILADELPHIA: 
CHRISTOPHER SOWER COMPANY, 

614 Arch Street. 



Copyright, 1S91, 
By EDWARD BROOKS. 



Westcott & Thomson, Sherman & Co. 

Sereotypers and Eleclrulypers, Phila. Printers, Phila. 



PREFACE. 



The little work on Plane Trigonometry, written by the 
Author and published, in connection with his Elementary 
Geometry, some twenty years ago, served to introduce the 
subject of Trigonometry into many schools not prepared 
to use the larger works on that subject. This work is still 
well adapted to the wants of many institutions, but other 
schools using the writer's series of mathematics desire a 
more complete work on the subject. 

To meet this demand, the present treatise has been pre- 
pared. It aims to furnish just so much of the subject as 
is taught in our best schools and colleges. Great care has 
been taken to give clearness and simplicity to the treat- 
ment, and to so grade the difficulties as to make the path- 
way of. the student smooth and easy. 

In treating the subject the method of ratios, now gen- 
erally adopted by mathematicians, has been employed. 
The old method of lines is presented in the latter part of 
the work, but care has been taken not to mix the two 
methods in the development of the principles. Some of 
the more difficult and less practical parts of the subject 
are printed in smaller type, and may be omitted by 



4 PREFACE. 

students desiring a shorter course. A large number of 
carefully graded problems are given to aid the student 
in fixing the principles and understanding their applica- 
tion. These exercises can be used at the option of the 
teacher and the requirements of the student. 

In preparing the work, the best American and English 
works on the science have been consulted, and some 
valuable material has been derived from Casey and Tod- 
hunter, especially in the Exercises for the application 
of the principles. 

Philadelphia, \ EDWAKD BEOOKS. 

May 10, 1891. J 



CONTENTS 



PAGE 

Introduction ........... 9 



PLANE TRIGONOMETRY. 

SECT. 

I. Measurement op Angles . . . . 23 

II. Trigonometrical Functions 27 

III. Functions op Angles in General .......... 36 

IV. The Sum and Difference of Two Angles 51 

V. The Theorems or Trigonometry ........... 59 

VI. Numerical Value of Sines, Tangents, etc 63 

' VII. The Solution of Triangles 72 

VIII. Practical Applications 85 

IX. Supplement ....... ....... 91 

SPHERICAL TRIGONOMETRY. 

X. Introductory Definitions ............. 107 

XL The Right Spherical Triangle 110 

XII. The Oblique Spherical Triangle 125 

XIII. Supplement * .144 

5 



HISTORY OF TRIGONOMETRY AND LOGARITHMS. 



Trigonometry is believed to have originated with the Greek 
astronomers of Alexandria. The foundations of the science seem 
to^have been laid by Hipparchus and Ptolemy. The first step in the 
science was the use of a table of chords, which served the same pur- 
pose as our table of sines. Ptolemy's celebrated work, the Almagest, 
contains a table of chords expressed in terms of the radius 5 and also 
the equivalent of several of our present formulas of trigonometry. 
Its treatment of spherical triangles was much more complete than 
that of plane triangles, which is natural, since the science was de- 
veloped in the interests of astronomy. 

The Indians at a very early date are known to have been familiar 
with the elements of the science, which they probably obtained from 
the Greeks. They introduced tables of half-chords, or sines, instead 
of chords, understood the relation between the sines and cosines of 
an arc and its complement, and could find the sine of half an* arc 
from the sine and cosine of the whole arc. 

The Arabs were acquainted with the Almagest, and probably 
learned from the Indians the use of the sine. Albategnius (930 a. d.) 
used the sine regularly, and was the first to calculate sin <f> from the 
formula sin -r cos <j> =*= k, He was acquainted with the formula cos a=* 
cos b cos c -j- sin b sin c cos A for a spherical triangle ABC. Abu 
l'Wafa of Bagdad (b. 940) was the first to introduce the tangent as 
an independent function. Gheber of Seville, in the 11th century, 
wrote an astronomy, the first book of which contains an article on 
trigonometry, much in advance of that of the Almagest. He gave 
proofs of the formulas for right spherical triangles, and presented 
for the first time the formulae cos B = cos b sin A, cos c = cot A 
cot B. He, however, made no advance in plane trigonometry. 

Johannes Miiller (1536-1476), known as Regiomontanus, wrote a 

treatise on the Almagest, in which he reinvented the tangent, and 

calculated a table of tangents for each degree, though he made no 

use of it and did not use formulas involving the tangent. This is 

6 



HISTORY OF TRIGONOMETRY AND LOGARITHMS. 7 

said to have been the first complete European treatise on trigo- 
nometry ; but its methods were in some respects behind those of the 
Arabs. Copernicus (1473-1543) gave the first simple demonstration 
of the fundamental formulae of spherical trigonometry. George 
Joachim, known as Rheticus (1514-1576), wrote a work which con- 
tains tables of sines, tangents, and secants of arcs at intervals of 
10 7/ from 0° to 90°. He found the formulae for the sines of the half 
and third of an angle in terms of the sine of the whole angle. 

Vieta (1540-1603) gave formulae for the chords of multiples of a 
given arc in terms of the chord of the simple arc. Albert Girard 
(1590-1634) published a work containing theorems which gave the 
areas of spherical triangles, and also employed the principle of sup- 
plementary triangles. He used the notation sin, tan, sec, for the sine, 
tangent, and secant of an arc. Newton gave the series for an arc in 
terms of its sine, from which he obtained the series for the sine and 
cosine in powers of the arc. James Gregory in 1670 discovered the 
series for the arc in powers of the tangent, and for the tangent and 
secant in powers of the arc. Leibnitz published in 1693 the series 
for the sine of an arc in powers of the arc. 

The greatest advance in the science was made by Euler (1707-1783), 
who really reduced the subject to its present condition. He intro- 
duced the present notation into general use, and made the transition 
from the geometrical conception of trigonometrical functions as lines 
to the analytical conception of functions of angles. The exponential 
values of sines and cosines, De Moivre's theorem, etc., are all due to 
Euler. 

History of Logarithms.— Logarithms were invented by Lord Na- 
pier, Baron of Merchiston, in Scotland. His first work upon- the sub- 
ject, entitled Mirifica Logarithorum Canonis, was published in 1614, 
and gave an account of the nature of logarithms, and a table of natural 
sines and their logarithms for every minute of the quadrant to seven 
and eight figures. A second work, published after Napier's death 
by his son in 1619, explained the method of constructing his table. 
These works did not contain the logarithms of numbers, but of sines 5 
and he called his numbers, not logarithms, but artificials. 

Napier's system of logarithms was afterward improved by Henry 
Briggs, a contemporary and friend of the inventor. Assuming 10 for 
the basis, he constructed a system of logarithms corresponding to our 
system of numeration, which is much more convenient forthe ordi- 
nary purposes of calculation. Briggs' s first work, a small octavo tract 
of 16 pages, was published in 1617, and contains the first published 



8 HISTORY OF TRIGONOMETRY AND LOGARITHMS. 

table of decimal or common logarithms. It gave the logarithms of 
numbers from unity to 1000 expressed to 14 places of decimals. A 
copy of the tract, now very rare, is found in the British Museum. 

In 1624, Briggs published a second work, entitled Arithmetical, Loga- 
rithmica, which contains the logarithms of numbers from 1 to 20,000, 
and from 90,000 to 100,000, calculated to 14 decimal places. In 1628, 
Adrian Vlacq, a native of Holland, published a work containing the 
logarithms of all numbers from 1 to 100,000. Vlacq' s table is that 
from which all the hundreds of tables since published have been 
derived. It contained many errors, which have gradually been dis- 
covered and corrected ; but, with one or two exceptions, no fresh 
calculations have ever been made. 

The first publication of the common logarithms of trigonometrical 
functions was made in 1620 by Gunter, a colleague of Briggs in 
Gresham College. This work contained logarithmic sines and tan- 
gents for every minute of the quadrant to 7 decimal places. In 1633, 
Vlacq published a work by Briggs, entitled Trigonometrica Britan- 
nica, which contained logarithmic sines and tangents at intervals of a 
hundredth of a degree. In the same year Vlacq published his Trigo- 
nometrica Artijicialis, giving logarithmic sines and tangents for every 
10 seconds of the quadrant to 10 decimal places. These were calcu- 
lated from the natural sines, etc., of the Opus Palatinum of Rheticus. 
This work fixed the method of applying logarithms to minutes and 
seconds, and it has never been superseded. 

Napier's system of logarithms is not now in use. A modification 
of this system is called the Napierian, or Hyperbolic, system. It is 
called Hyperbolic because the logarithms represent the area of a 
rectangular hyperbola between its asymptotes. The base of the 
Napierian system is 2.718, and is denoted by the letter e. The first 
logarithms to the base e were published by John Speidell in 1619, 
in a work entitled Neiv Logarithms. It contains hyperbolic log. sines, 
etc., for every minute of the quadrant to 5 places of decimals. 

For information on centesimal logarithms, airtilogarithms, logistic 
numbers, Gaussian logarithms, etc., see Enci/clopcedia Britannica, 
from which most of the above history is collated. 



INTRODUCTION". 



THE LOGARITHMS OF NUMBERS. 

1. Logarithms are a species of numbers used to abbreviate 
multiplication, division, involution, and evolution. 

2. The Logarithm of a number is the exponent denoting the 
power to which a fixed number must be raised to produce the first 
number. 

Thus, if B x = jV, then x is called the logarithm of N. 

3. The Base of the system is the fixed number which is raised to 
the different powers to produce the numbers. 

Thus, in B x ~ iV, x is the logarithm of iVto the base jB; so in 
4 3 = 64, 3 is the logarithm of 64 to the base 4. 

4. The term logarithm, for convenience, is usually written log. 
The expressions above may be written log N= x ; and log 64 = 3. 

5. In the Common System of logarithms the base is 10, and the 
nature of logarithms is readily seen with this base. Thus, 

10 1 =10; hence log 10 =1. 

10 2 = 100; hence log 100 = 2. 

10 3 = 1000 ; hence log 1000 = 3. 
10 2 - 369 = 234; hence log 234 =2.369. 

G. We shall first derive the general principles of logarithms, the 
base being any number, and then explain the common numerical 
system. 

Principles of Logarithms, 

Prin. 1. The logarithm of 1 is 0, whatever the base. 

For, let B represent any base, then B° = 1 ; hence by the definition of 
a logarithm, is the log of 1, or log 1 = 0. 



10 INTRODUCTION. 

Prin. 2. The logarithm of the base of a system of logarithms is 
unity. 

For, let B represent any base, then B 1 = B ; hence 1 is the log of B, 
or log B = 1. 

Prin. 3. The logarithm of the product of two or more numbers is 
equal to the sum of the logarithms of those numbers. 

For, let m = log M, and n = log N. 

Then, 5« = M, B™ = N. 

Multiplying, B™ + n = i!/" X A r . 

Hence, m -}- ?i = log (if X N). 

Or, log {M X N) = log if + log N. 

Prin. 4. The logarithm of the quotient of two numbers is equal to 
the logarithm of the dividend minus the logarithm of the divisor. 

For, let m = log M, and n = log N. 

Then, B™ = M, B" = JV. 

Dividing, B™ ~ » = M — iV. 

Hence, log (if 4* i\ 7 ) = m — «. 

Or, log (if -r- N) =■ log if— log N. 

Prin. 5. I%e logarithm of any power of a number is equal to the 
logarithm of the number multiplied by the exponent of the power. 

For, let m = log M. 

Then, B™ = M. 

Raising to nth power, B*> x m — if n 
Whence, log if M = n X m. 

Or, log ifw — n X log Jf. 

Prin. 6. The logarithm of the root of a number is equal to the 
logarithm of the number divided by the index of the root. 

For, let m = log M. 

Then, B™ = M. 

m l 

Taking nth root, B n = 3P 1 . 

"Whence, log M n — — 



Or, log if* 



n 

locr M 



INTE OD UCTION. . 1 1 

7. These principles are illustrated by the following exercises, 
which the pupil will work. 

EXERCISES. 

Prove each of the following: 

1. Log (a.b.c.) ■= log a 4* log b + log c. 

2. Log { — j = log a 4- log b — - log c. 

3. Log a n = n log a, 

4. Log {a x bv) = x log a -f- y log 6. 



5. Log — — = x log a-\-y log & — z log c. 

6. Log Vab = \ log a -f- £ log 6. 

7. Log (a 2 — a; 2 ) = log (a + a;) -}- log (a — #). 



8. Log ]/a 2 — x 2 = £ log (a + #) + J log (a — *)• 

9. Log a 2 |/a -2 = f log a. • 



10. Log ^--t^ = J {log (a -a:) -3 log (a 4- *)i. 
(a 4- a)- 

Common Logarithms. 

8. In Common Logarithms the base is 10. This base is most 
convenient for numerical calculations, because our numerical system 
is decimal. 

9. In this system every number is conceived to be some power of 
10, and by the use of fractional and negative exponents may be thus, 
approximately, expressed. 

10. Raising 10 to different powers, we have — 

hence = log 1. 
hence 1 = log 10. 
hence 2 = log 100. 
hence 3 = log 1000. 
etc. 

hence — 1 = log .1. 

hence — 2 = log .01. 
001 ; hence — 3 = log .001. 





10° = 


i; 




10! = 


10; 




10 2 = 


100; 




10 3 = 


1000 




etc 




Also, 


10- 1 = 


.1; 




io- 2 = 


.01; 




io- 3 = 


.001 



1 2 INTR OB UCTION. 



11. Hence the logarithms of all numbers 

between 1 and 10 will be + a fraction $ 
between 10 and 100 will be 1 -|- a fraction ; 
between 100 and 1000 will be 2 -f- a fraction ; 
between 1 and .1 will be — 1 + a fraction ; 
between .1 and .01 will be — 2 -f a fraction ; 
between .01 and .001 will be — 3 -f- a fraction. 

12. Thus it has been found that the log of 76 is 1.8808, and the 
log of 458 is 2.6608. This means that 

10 1 - 8808 = 76, and 10 2 - 6608 = 458. 

13. When the logarithm consists of an integer and a decimal, the 
integer is called the characteristic, and the decimal part the mantissa. 
Thus, in 2.660865, 2 is the characteristic, and .660865 is the mantissa. 

14. The logarithm of a number less than 1 is negative ; but is 
written in such a form that the fractional part is always positive. 
Thus, log .008 = log (8 X .001) = log 8 -f log .001 = 0.903090 — 3. 
Now, this may be written 3.903090. The minus sign is written over 
the characteristic to show that it only is negative. 

Principles of Common Logarithms. 

Prin. 1 . The characteristic of the logarithm of a number is one less 
than the number of integral places in the number. 

For, from Art. 10, log 1 =0 and log 10 = 1 ; hence the logarithm 
of numbers from 1 to 10 (which consist of one integral place) will 
have for the characteristic. Since log 10 = 1 and log 100 = 2, the 
logarithm of numbers from 10 to 100 (which consist of two integral 
places) will have one for the characteristic, and so on ; hence the 
characteristic is always one less than the number of integral places. 

Prin. 2. The characteristic of the logarithm of a decimal is neg- 
ative, and is equal to the number of the place occupied by the first 
significant figure of the decimal. 

For, from Art. 10, log .1 =— 1, log .01 = — 2, log .001 = — 3 ; 
hence the logarithms of numbers from .1 to 1 will have — 1 for a 
characteristic ; the logarithms of numbers between .01 and .1 will 



INTR OD UCTION. 1 3 

have — 2 for a characteristic, and so on ; hence the characteristic of 
a decimal is always negative, and equal to the number of the place of 
the first significant figure of the decimal. 

Prin. 3. The logarithm of the product of any number multiplied by 
10 is equal to the logarithm of the number increased by 1. 

For, suppose log M = m • then, by Prin. 3, Art. 6, 

log (MX 10) = log M + log 10 ; but log 10 == 1 ; 
Hence, log (M X 10) = m -f 1 . 
Thus, log (76 X 10) = 1.880814 -f 1 ; or log 760 = 2.880814. 

Prin. 4. The logarithm of the quotient of any number divided by 
10 is equal to the logarithm of the number diminished by 1. 

For, suppose log M= m ; then, by Prin. 4, Art. 6, 

log (M~ 10) = log M— log 10 ; 
Hence, log (M— 10) = m — 1. 
Thus, log (458 -f- 10) = 2.660865 — 1 ; or log 45.8 = 1.660865. 

Prin. 5. In changing the decimal point of a number we change the 
characteristic, but do not change the mantissa of its logarithm. 

This follows from Principles 3 and 4. To illustrate : 

log 234 = 2.369216. log .234=1.369216. 

log 23.4 = 1.369216. log .0234 = 2.369216. 

log 2.34 = 0.369216. 

We thus see that when we change the place of the decimal point of a 
number we change the characteristic, but do not change the decimal part 
of the logarithm. 

15. Negative logarithms are sometimes written with 10 or a mul- 
tiple of 10 after them, and a positive characteristic equal to the differ- 
ence between its real characteristic and 10 or the given multiple 
of 10. 

Thus, 2.369216 may be written 8.369216 — 10; and 13.369216 
may be written 7.369216 — 20. 



14 INTRODUCTION. 

Table of Logarithms. 

16. A Table of Logarithms is a table by means of which we 
can find the logarithms of numbers, or the numbers corresponding 
to given logarithms. 

17. In the annexed table the entire logarithms of the numbers up 
to 100 are given. For numbers greater than 100 the mantissa alone 
is given ; the characteristic being found by Prin. 1, page 12. 

18. The numbers are placed in the column on the left, headed N ; 
their logarithms are opposite, on the same line. The first two figures 
of the mantissa are found in the first column of logarithms. 

19. The column headed D shows the average differences of the 
ten logarithms in the same horizontal line. This difference is found 
by subtracting the logarithm in column 4 from that in column 5, and 
is very nearly the mean or average difference. 

Note. — The logarithms given in the Table are complete to six places. 
They can be readily changed to five-place logarithms by omitting the 
sixth figure, and when the sixth figure is 5 or more, increasing the fifth 
figure by 1. Similarly, we find four-place and three-place logarithms. 

To Find the Logarithm of any Number. 

20. To find the logarithm of a number o/*one or two figures. 

Look on the first page of the table, in the column headed N, and 
opposite the given number will be found its logarithm. Thus, 

the logarithm of 25 is 1.397940, . 
" " 87 is 1.939519. 

21. To find the logarithm ofi a number of three figures. 

Look in the table for the given number ; opposite this, in column 
headed 0, will be found the decimal part of the logarithm, to which 
we prefix the characteristic 2, Prin. 1. Thus, 

the logarithm of 325 is 2.511883, 
" " 876 is 2.942504. 

22. To find the logarithm of a number of FOtrR figures. 

Find the three left-hand figures in the column headed N, and 
opposite to these, in the column headed by the fourth figure, will be 



INTRODUCTION. 15 

found four figures of the logarithm, to which two figures from the 
column headed are to be prefixed. The characteristic is 3, Prin. 1. 
Thus, 

the logarithm of 3456 is 3.538574, 
" " 7433 is 3.871456. 

23. In some of the columns small dots are found in the place of 
figures : these dots mean zeros, and should be written zeros. If the 
four figures of the logarithm fall where zeros occur, or if, in passing 
back from the four figures found to the zero column, any of these 
dots are passed over, the two figures to be prefixed must be taken 
from the line just below. Thus, 

the logarithm of 1738 is 3.240050, 
" " 2638 is 3.421275. 

24. To find the logarithm of a number of more than four figures. 

Place a decimal point after the fourth figure from the left hand, 
thus changing the number into an integer and a decimal. Find the 
mantissa of the entire part by the method just given. Then from 
the column headed D take the corresponding tabular difference, mul- 
tiply it by the decimal part, and add the product to the mantissa 
already found ; the result will be the mantissa of the given number. 
The characteristic is determined by Prin. 1. 

If the decimal part of the product exceeds .5, we add 1 to the 
entire part ; if less than .5, it is omitted. 



EXERCISES. 

1. Find the logarithm of 234567. 

Solution. — The characteristic is 5, Prin. 1. Placing a decimal 
point after the fourth figure from the left, we have 2345.67. The 
decimal part of the logarithm of 2345 is .370143; the number in 
column D is 185 ; and 185 X .67 = 123.95, and since .95 exceeds .5, 
we have 124, which, added to .370143, gives .370267 ; hence, log 
234567 = 5.370267. 



16 



INTRODUCTION. 



Find the logarithm 



2. 


Of 4567. 


Ans. 


3.659631. 


8. 


Of 704.307. 


Ans. 


2.847762. 


3. 


Of 3586. 


Ans. 


3.5546l5. 


9. 


Of .000476. 


Ans. 


4.677607. 


4. 
5. 


Of 11806. 
Of .4729. 


Ans. 
Ans. 


4.072102. 
T.674769. 


10. 


~.r. 365 
400 ' 


Ans. 


1.960233. 


6. 

7. 


Of 29.337. 
Of 734582, 


Ans. 
Ans. 


1.467416. 
5.866040. 


11. 


Of 515.- 

463 


Ans. 


"1.908450. 



Note.— To find the logarithm of a common fraction, subtract the log- 
arithm of the denominator from the logarithm of the numerator. 

25. To find the number corresponding to a given logarithm. 

1. Find the two left-hand figures of the mantissa in the column 
headed 0, and the other four, if possible, in the same or some other 
column on the same line ; then, in column N, opposite to these latter 
figures, will be found the three left-hand figures, and at the top of the 
page the other figure of the required number. 

2. When the exact mantissa is not given in the table, take out the 
four figures corresponding to the next less mantissa in the table ; sub- 
tract this mantissa from the given one ; divide the remainder, with 
ciphers annexed, by the number in column D, and annex the quo- 
tient to the four figures already found. 

3. Make the number thus obtained correspond with the character- 
istic of the given logarithm, by pointing off decimals or annexing 
ciphers. 

EXERCISES. 

1. Find the number whose logarithm is 5.370267. 

Solution. — The mantissa of the given logarithm is . . .370267 
The mantissa of the next less logarithm of the table is . . .370143 

and its corresponding number is 2345. 

Their difference is , . . 124 

The tabular difference is 185 

The quotient is 185)124.00(.67 

Hence the required number is 234567 

Note. — If the characteristichad been 2, the number would have been 
234.567 ; if it had been 7, the number would have been 23456700 ; if it had 
been 2, the number would have been .0234567, etc. 



INTRODUCTION. 



17 



Find the number whose logarithm 



2. 


Is 3.659631. 


AllS. 


4567. 


3. 


Is 3.563125. 


Ans. 


3657. 


4. 


Is 2.554610. 


Ans. 


358.6. 


5. 


Is 1.072102. 


Ans. 


11.806 


6. 


Is 4.883150. 


Ans. 


76410. 



7. Is 4.790285. Ans. 61700. 

8. Is 2.674769. Ans. .04729. 

9. Is 3.065463. Ans. .0011627. 

10. Is 3.514548. Ans. .00327. 

11. Is 4.846741. Ans. .00070265. 



Multiplication by Logarithms. 

26. From Prin. 3, for the multiplication of numbers by means of 
logarithms, we have the following 

Rule. — Find the logarithms of the factors, take their sum, and find 
the number corresponding to the result; this number will be the re- 
quired product. 

Note. — /The term sum is used in its algebraic sense. Hence, when any 
of the characteristics are negative, we take the difference between the 
sums of the positive and negative characteristics, and prefix to it the 
sign of the greater. If anything is to be " carried " from the addition of 
the mantissa, it must be added to a positive characteristic or subtracted 
from a negative one. 

When any of the characteristics are negative, we can write them as 
suggested in Art. 15, aud proceed accordingly. 



EXERCISES. 



1. Multiply 35.16 by .815. 
Solution. — 



log 35.16 
log .815 



28.6554 



1.546049 
T.911158 
1.457207 

457125 

152)82.00(.54 



Product, 

Find the product 

2. Of .7856, 31.42. Ans. 24.6835. 

3. Of 0.3854 by 0.0576. Ans. .022199. 

4. Of 31.42, 56.13, and 516.78. Ans. 911393.7. 

5. Of 31.462, .05673, and .006785. Ans. .01211168. 

6. Of .06517, 2.16725, .000317, and 42.1234. Aits. .001886. 

7. Of 2.3456, .00314, 123.789, .00078, and 67.105. Ans. .04772076. 

2 



18 



INTRODUCTION. 



Division by Logarithms. 

27. From Priii. 4, to divide by means of logarithms, we have the 
following 

Rule. — Find the logarithms of the dividend and divisor, subtract 
the latter from the former, and find the number corresponding to the 
result; this number ivill be the required quotient. 

Note. — The term subtract is here used in its algebraic sense : hence, 
when any of the characteristics are negative we mast subtract according 
to the principles of algebra. 

Negative characteristics may be written as in Art. 15, for subtraction. 

EXERCISES. 

1. Divide 783.5 by .625. 
First Solution. 
log 783.5 = 2.894039 
log .625=1.795880 



3.098159 
Quo. 1253.6 .097951 

346)208(6 

2. Divide 272.636 by 6.37. 

3. Divide 50.38218' by 67.8. 

4. Divide 155 by .0625. 

5. Divide 1.1134 by 0.225. 

6. Divide 0.10071 by 0.00373. 

7. Divide 435 X 684 by 583 X 760. 



Second Solution. 
log 783.5 = 12.894039 — 10 
log .625= 9.795880 — 10 

Dif. 



3.098159 



Quo. 1253.6. 



Ans. 42.8. 

Ans. .7431. 

Ans. 2480. 

Ans. 5.04. 

Ans. 27. 

Ans. 671524. 



The Cologarithm of a Number. 

28. The Cologarithm of a number is the result arising from sub- 
tracting the logarithm of the number from 10. Thus, colog N= 10 
— log N, and colog 40 = 10 — log 40, or 10— 1.60206 = 8.39794. 

29. The cologarithm of a number may be written directly from 
the table by subtracting each term of the logarithm from 9, except 
the right-hand term, which must be taken from 10. 

30. The cologarithm is used to simplify the operation of division 



INTE OD UCTION. 1 9 

when it is combined with multiplication. Thus, suppose we wish to 
divide M by N. 

Now, log (Jf -f- N) = log M — log N. 

But, log K= 10 — colog K. Art. 28. 

Substituting, log M — log A^— log M + colog N — 10. 

Hence, instead of subtracting log -A 7 , we may add colog iV, and 
then deduct 10 from the sum. 

31. Hence, to divide by means of the cologarithm of a number 

we have the following 
i 

Rule. — Add the cologarithm, of 1 the divisor to the logarithm of the 

dividend, subtract 10, and jind the number corresponding to the 

result. 

Note. — The cologarithm is sometimes defined as the logarithm of the 
reciprocal of the number, and the rule for its use deduced accordingly. 
The cologarithm as defined above is usually known as the Arithmetical 
Complement. 

EXERCISES. 

1. Divide 256.3 by 45.32. 

Solution.— log 856.3 2.932626 

colog 45.32 8.343710 

Quotient, 18.8945 1.276336 

2. Divide 0.3156 by 78.35. 

log 0.3156 1.499137 

colog 78.35 8.105961 

Quotient, .004028 3.605098 

3. Divide 3.7521 by 18.346. Ans. .204519. 

4. Divide 483.72 by .30751. Ans. 1573.02. 

5. Find value of 32.16 X 7.856 ~ 45.327. Ans. 5.574. 

6. Of 31.57 X 123.4 divided by 316.2 x .0316. Ans. 389.8884. 

7. Of ar, given x : 73.15 = 40.16 : 3167. Ans. 1.11237. 

8. Of x, given 72.34 : 2.519 = 357.48 : x. Ans. 12.448. 



20 INTR OD UCTION. 



Involution by Logarithms. 
32. From Prin. 5, to raise a number to any power, we have the 
following 

Rule. — Find the logarithm of the number, multiply it by the expo- 
nent of the power, and find the number corresponding to the result. 

EXERCISES. 

1. Find the 4th power of 45. 
Solution.— log 45 = 1.653213 

4 



Power, 4100625 6.612852 

2. Find the cube of 0.65. Ans. 0.2746. 

3. Find the 6th power of 1.037. Ans. 1.243. 

4. Find the 7th power of .4797. Ans. 0.005846. 

5. Find the 30th power of 1.07. Ans. 7.6123. 

Evolution by Logarithms. 

33. From Prin. 6, to extract from any root of a number, we have 
the following 

Rule. — Find the logarithm, of the number, divide it by the index 
of the root, and find the number corresponding to the result. 

Note. — If the characteristic is negative, and not divisible by the 
index of the root, add to it the smallest negative number that will make 
it divisible, prefixing the same number with a plus sign to the mantissa. 

EXERCISES. 

1. Find the square root of 576. 

Solution.— log 576 = 2.760422 

2.760422-^2 = 1.380211 
Hence, the root is 24. 

2. Find the fourth root of .325. 
Solution.— log .325 =1.511883 

— 3 +3 

4) _ 4 _l. 3.511883 

1.877971 . 
Hence, the root is, .75504. 



INTRODUCTION. 21 

3. Find the cube root of 7. Ans. 1.9129. 

4. Find the fifth root of 5. Ans. 1.3797. 

5. Find the fifth root of .0625. Ans. .57434-8. 

6. Find the seventh root of 7. Ans. 1.32047. 

7. Find the tenth root of 8764.5. Ans. 2.479. 

Calculation of Logarithms. 

The pupil will naturally desire to know how these logarithms are 
calculated. While this is not the place to enter into a detailed ex- 
planation of the method of calculating logarithms, a general idea of 
the subject can be presented. 

In computing logarithms it is necessary to calculate only the log- 
arithms of prime numbers, since the logarithms of composite numbers 
may be obtained by adding the logarithms of their prime factors. 

The logarithms of the prime numbers were first computed by com- 
paring the geometrical and arithmetical series, 1, 10, 100, etc., and 0, 
1, 2, etc., and finding geometrical and arithmetical means ; the arith- 
metical mean being the logarithm of the corresponding geometrical 
mean. This method was exceedingly laborious, involving so many 
multiplications and extractions of roots. 

The method now generally used is that of series, by which the 
computations are much more easily made. The following formula is 
derived by algebraic reasoning : 

(<-Y» rwtit /yii> /y»* /V«U \ 

f-r+f- f-+r- et 4 

In this series the quantity A is called the modulus, which in the 
Napierian system is unity. The series, when A is one, put in a 
more convenient form, becomes 

l„ g ( z+ l)-lo g « = 2(^ + ^L__ + ^i IF + etc.). 

From which, knowing the logarithm of any number, we readily 
find the logarithm of the next larger number. The student will be 
interested in finding logarithms by this formula. Begin with 2, in 
which 8 = 1. 



22 INTRODUCTION. 

The logarithm found will be the Napierian logarithm, and this 
multiplied by 0.434294 will give the common logarithm. 

Logarithms were invented by Lord Napier of Scotland, and are 
regarded as among the most useful inventions ever made. His sys- 
tem was subsequently improved by Henry Briggs, a cotemporary of 
Napier's, who, assuming 10 for a basis, constructed a system much 
more convenient for the ordinary purposes of computation. Napier's 
system was also modified by John Speidell, whose logarithms are 
now known as the Napierian or Hyperbolic logarithms. Briggs' 
logarithms are known as the Briggean or Common logarithms. 

It is generally believed that the so-called " Napierian logarithms" 
are identical with those first computed by Napier ; but this is not the 
case. For a more detailed statement of the origin of logarithms, see 
the History of Logarithms given in the Introduction of this work, 
page 6. 



PLANE TRIGONOMETRY. 



SECTION I. 

THE MEASUREMENT OF ANGLES. 

1. Trigonometry is the science which investigates the 
relation of the sides and angles of triangles. 

2. Plane Trigonometry treats of plane angles and tri- 
angles ; Spherical Trigonometry treats of spherical angles 
and triangles. 

3. In every triangle there are six parts, three sides and 
three angles. These parts are so related that when certain 
ones are given, the others may be found. 

4. In Geometry the triangle can be constructed when a 
sufficient number of parts are given. In Trigonometry the 
unknown parts are computed from the known parts. 

5. In order to subject a triangle to computation, we must 
be able to express its sides and angles by numbers. For 
this purpose proper units must be adopted. 

6. The units of measure for the sides are straight lines 
of a fixed length, as the inch, foot, yard, etc. The units of 
measure for angles are degrees, minutes, and seconds. 

Note. — Trigonometry is really a numerical way of treating trian- 
gles in distinction from the geometrical way of treating them. The 
science extends also to the investigation of angles in general, and is 
then called Angular Analysis. 

23 



24 



PLANE TRIGONOMETRY. 



Measures of Angles. 
7. An angle is measured, as shown in geometry, by the 
arc intercepted between its sides, the centre of the circle 
being at the vertex of the angle. 

S. The units of the arc are equal parts of the circum- 
ference called degrees, minutes, and seconds. A degree 
(marked °) is -^^ of the circumference ; a minute (marked ') 
is -gV of a degree ; and a second (marked ") is^ -£$ of a 
minute. 

9. A Quadrant is one-fourth of the circumference of a 
circle. Each quadrant contains 90°, and is the measure 
of a right angle. 

10. Reckoning from A, the arc AB 
is called the first quadrant; the arc BC 
the second quadrant; the arc CD the 
third quadrant ; the arc DA the fourth 
quadrant. The term quadrant is also 
applied in the same manner to the four 
equal parts of the circle. 

11. Any arc AE, less than 90°, is said to be in' the first 
quadrant ; any arc AF, between 90° and 180°, is said to be 
in the second quadrant; any arc A G, between 180° and 
270°, in the third quadrant, etc. 

12. The Complement k of an angle or an arc is the re- 
mainder obtained by subtracting the angle or arc from 90°. 
Thus, the complement of arc AE is arc BE. 

13. The Supplement of an angle or an arc is the re- 
mainder obtained by subtracting the angle or arc from 
180°. Thus, the supplement of arc AF is arc CF. 




Fig. l. 



NUMERICAL LENGTHS OF ARCS. 25 

14. According to these definitions, the complement of an 
arc greater than 90° is negative, and the supplement of an 
arc greater than 180° is negative. Thus, the complement 
of 120° is 90° — 120° = — 30° ; and the supplement of 
200° is 180° - 200° = - 20°. 

Numerical Lengths of Arcs. 

15. The units of the circle — that is, degrees, minutes, 
and seconds — express equal parts of the circumference. 
An arc may also be expressed in numerical units cor- 
responding to a straight line. 

I. To find a numerical expression for an arc of a given num- 
ber of degrees, minutes, etc. 

The circumference of a circle is 2ttR (B. V., Th. 8). 
Supposing R = 1, we find the semi-circumference equal to 
n = 3.14159265. Hence, 

Arc 180° = 3.14159265. Arc 1' = 0.000290888. 

Arc 1° = 0.01745329. Arc 1" = 0.000004848. 

II. To find the number of degrees, minutes, etc., in, an arc 
equal to the radius. 

Since, 2nR = 360°, tt R = 180°. 

1 80° 1 S0° 

Hence, R = ±^L = _^L _ 57°. 2 957795. 
tt 3.1415926o 



= 3437'.74677= 206264". 
16. The angle at the centre measured by an arc equal to 
the radius, it is thus seen, is an invariable angle, whatever 
the length of the radius ; hence it is often taken as the unit 
of angular measure. • 

IT. Since when the radius is unity, 2n = 360°, tt is often 
used to express two right angles. Then - equals a right 



26 PLANE TRIGONOMETRY. 

angle; 2tt equals four right angles; - — an angle of 
45°, etc. 

18. This method of measuring an angle is called the cir- 
cular measure of an angle. The method by degrees, etc. is 
called the sexagesimal method. 

Note. — A third method, called the centesimal method, was pro- 
posed by the French at the introduction of the metric system. In 
this system the right angle was divided into 100 parts, called grades, 
each grade into 100 parts called minutes, etc. 

EXERCISES I. 

1. How many degrees in an angle denoted by27r? By it? By 
£tt? By37r? ByfTr? By^Tr? By^-Tr? nir? 

2. Express in terms of -k an angle of 180° ; of 90° ; of 60° ; of 45° ; 
of 30° ; of 70° ; of 80° ; of 63° ; of 67° 30' ; of 52° 30'. 

3. How many degrees in an arc whose length is equal to the diam- 
eter of the circle ? Ans. 114°. 59 -f . 

4. How many degrees in an arc -whose length is 0.6684031 ? Whose 
length is 2.0052093 ? Ans. 38° 17' 48" ; 114° 53' 24". 

5. Express -^ of a right angle in degrees and minutes ; also in cir- 
cular measure. Ans. 28° 7 / .5 ; -^tt. 

6. What is the length of an arc of 60° when the radius is 8 ? 
When the radius is 12 ? Kadjus 20 ? 

7. When the radius is 8, required the length of an arc of 45° ; of 
75°; of 22° 30 / ; of 52° 30' ; of 33° 45'. 

8. Find the diameter of a globe when an arc of a great circle of 
25° measures 4 feet. Ans. 18.3346. 

9. Find the number of degrees in a circular arc 30 inches in length, 
the radius being 25 inches. * Ans. 68° 45 / 17" +• 



TRIGONOMETRICAL FUNCTIONS. 



27 



SECTION II. 



TRIGONOMETRICAL FUNCTIONS. 

19. In Trigonometry, instead of comparing the angles 
of triangles or the arcs which measure them, we compare 
certain lines or ratios of lines called the functions of the 
angles. 

20. A Function of a quantity is something depending on 
the quantity for its value. These functions in Trigonometry 
are the sine, cosine, tangent, cotangent, secant, and cosecant 

21. Since every oblique triangle can be resolved into two 
right triangles by drawing a perpendicular from one of its 
angles to the opposite side, the solution of all triangles can 
be made to depend on that of right triangles. 

22. The functions, sine, cosine, etc., are used to express 
the relation of the sides of the right triangle. These terms 
will now be denned and illustrated. 

23. In the right triangle ABC, let 
AC be denoted by o, BC by a, and AB 
by c ; then we have the following defi- 
nitions : 

1. The Sine of an angle is the ratio of 
the opposite side to the hypotenuse. 

_ . , BC a . n AC b 

Thus, sin A = -jt: — ~ \ sin B = —rz: — - • 
AB c ' AB c 

2. The Tangent of an angle is the ratio of the opposite side 
to the adjacent side. 

Thus, tan A = -j^ = y, tan B = -^ = -• 




28 PLANE TRIGONOMETRY. 

3. The Secant of an angle is the ratio of the hypotenuse 
to the adjacent side. 

mi . AB c n AB c 

Thus, sec^ = — =-;sec5=^=-. 

4. The Cosine of an angle is the sine of the complement of 
the angle. 

Thus, cos A = sin B = - ; hence cos A — - • 

c ' c 

Also, cos B = sin A = - ; hence cos B = - • 

c ' c 

5. T/ie Cotangent of an angle is the tangent of the comple- 
ment of the angle. 

Thus, cot A — tan B — - ; cot B = tan J = 7 • 

a' b 

6. T/*e Cosecant of an angle is the secant of the complement 

of the angle. 

c c 

Thus, esc A = sec B — - ; esc i? = sec A = - • 

24. If ^4 denotes any angle or arc, then we have from the 
above explanations, 

sin A = cos (90° —A)] cos A = sin (90° - A). 

tan A = cot (90° — A) ; cot ^ = tan (90° - A). 

sec A = esc (90° — J); esc A = sec (90° - A). 

Note. — The above definitions of cosine, cotangent, and cosecant 

show their true relation to the sine, tangent, and secant* We may, 

however, define them independently, as follows : 

1. The cosine of an angle is the ratio of the adjacent side to the 
hypotenuse. 

2. The cotangent of an angle is the ratio of the adjacent side to the 
opposite side. 

3. The cosecant of an angle is the ratio of the hypotenuse to the 
opposite side. 



TRIGONOMETRICAL FUNCTIONS. 29 

25. The sine, cosine, tangent, cotangent, etc. are called 
Trigonometrical Functions or Ratios. A large part of Trigo- 
nometry consists in the investigation of the properties and 
relations of these functions of an angle. 

Note. — If the cosin| of an angle is subtracted from unity, the re- 
mainder is called the versed-sine of an angle ; if the sine of an angle 
is subtracted from unity, the remainder is called the coversed-sine of 
the angle. 

Thus, vers A = 1 — cos A ; covers A = 1 — sin J.. 

EXERCISES II. 

1. Find the values of the trigonometrical functions of A, when 
a = 3, 6 = 4, and c = 5. 

Solution.— By Art. 23, 

, a 3 , 6 4 , . a 3 , 

sin A = - = - : cos A = - = - ; tan A = _ = — ; etc. 
c 5 ' c 5 ' 6 4 

2. Find the values of the trigonometrical' functions of A when 
a = 5, 6 = 12, and c = 13. When a = 8, b — 15, and c = 17. 

3. Write all the functions of B in the triangle of Fijr. 2. 

4. Find the functions of A and of B when a = 10 and 6 = 24. 
When a = 18 and c = 82. When 6 = 75 and c = 85. 

5. Find the functions of A and of B when a = m and c = m j/2. 
When 6 = |/2;»7i and c == m -J- w. 

6. Find a, if sin ^1 = f and c = 5. Find 6, if tan A = ^ and 
c = 13. Find c, if sec ^4 = 2 and 6 = 5. 

7. Compute the functions of J. when a = f 6. When 6 = f c. When 
a 4. 6 = f c. When a — 6 = \c. 

8. Construct a right triangle when sin A=^ and a = 9. When 
tan A = | and 6 = 9. When esc A = 3.5 and c = 4|. 

9. Compute the legs of a right triangle when sin A = 0.4, cos 
A = 0.6, and c = 4.5. Construct the triangle. 

10. Given A + B = 90° ; to prove the following : 

sin A = cos B. tan J[ = cot B. sec ^4 = esc B. 

cos J. = sin B. cot ^4 = tan B. esc J. = sec B. 




30 PLANE TRIGONOMETRY. 

Fundamental Formulas. 
26. We now proceed to derive some fundamental form- 
ulas expressing the relations of trigonometrical functions. 

I. Formulas expressing the relation of sine and 
cosine. 

1. In the right triangle ABC, by 
geometry, 

We have, a 2 + b 2 = c 2 . 

Whence, - + - = 1. p . g 3 

Substituting the values of sin A and cos A (Art. 23), 
We have, sin 2 A + cos 2 A = 1. [1] 

That is : The sum of the squares of the sine and cosine of an 
angle is equal to unity. 

Note. — We write sin 2 A for (sin A) 2 and cos 2 A for (cos A)' 2 as a 
matter of convenience, and similarly with the powers of the other 
trigonometrical functions. 

2. From the above formula we have 

sin 2 A — 1 — cos 2 A = (1 + cos ^4) (1 — cos ^4). 
cos 2 A== 1 — sin 2 A = (1 + sin A) (1 — sin A). 

II. Formulas expressing the relation of tangent 
and cotangent. 

1. From Art. 23, (1) and (4), we have 

sin A a o a , . a . , 00 /nN 

: = - -~ - — t ; but tan ^4 = 7 • Art. 23, (2). 

cos ^4 c c b ' & 

Whence, tan A = • [2] 

cosA J 

That is : The tangent of an angle is equal to the sine divided 
by the cosine. 



FUNDAMENTAL FORMULAS. 31 

2. From Art. 23, (5), we have 

, '. b . cos A b a b 

cot A = - ; but — t = - -r- - = - • 

a 5 sini c c a 

Whence, cot A =^|. [3] 

sin A L J 

That is : The cotangent of an angle is equal to the cosine 

divided by the sine. 

3. Taking the product of [2] and [3], we have 

tan A x cot A = 1. 

Whence, tan A = — — r- ; or cotA=- — • [4] 

That is : The tangent and the cotangent of an angle are re- 
ciprocals of each other. 

III. Formulas expressing the relation of secant 
and cosecant to the other functions. 

ft f> 

1. From Art. 23, sin A = - and esc A = - • 

c a 

Whence, sin A = ; or esc A = — — r • RH 

esc A ' sm A L J 

That is : The sine and cosecant of an angle are reciprocals 

of each other. 

b c 

2. From Art. 23, cos A = - and sec A = t • 

. c b 

Hence, cos A = and sec A = • r61 

sec A cos A 

That is : The cosine and secant of an angle are reciprocals 

of each other. 

3. In the right triangle ABC 
We have c 2 = b 2 + a 2 . 

Dividing by b 2 , ~ 1 + ^- 

Whence, sec 2 A = 1 + tan 2 A. [7] 



32 



PLANE TRIGONOMETRY. 



In a similar manner we find 

esc 2 A = 1 4- cot 2 A. 

27. These expressions derived under Art. 26 may be 
regarded as the Fundamental Formulas of Trigonometry, 
and should be committed to memory. We shall collect 
them, forming the following table: 

Table I. 



1. SinM + cos 2 J. 

2. Sin 2 A 

3. Cos 2 A 

4. Tan A = 

5. Cot A = 

6. Tan A cot A = 

7. Tan A = 



1 — cos 2 A ■ 

1 — sin 2 A 

sin A 



cos 


A 


COS 


A 


sin 


A 


1 




I 





cot A 



tan 


i 


1 




COS 


A 


1 





8. Cot A = 

9. Sec J. 

10. Csc A = 

sin A 

11. Sec 2 ^l = 1 + tan'M 

12. Csc 2 A =l + cot 2 ^ 

13. Ver sin J. =1 — cos A 

14. Co-vcr sin A =1 — sin A 



Note. — 1. The student should be able to state these formulas and 
also express them in theorems. 

2. The student should also fix the following truths in his under- 
standing : 

(a) Either side of a right triangle equals the hypotenuse into the 
sine of the opposite angle. 

(b) Either side of a right triangle equals the hypotenuse into the 
cosine of the adjacent angle. 

EXERCISES III. 

Find the values of the other functions, when 

1. sinj; = f. 5. tan A=^. 9. sec .4= f. 

2. sin A = T 5 3. 6. tan A = 2. 10. csc A == -j/5. 



3. cos A = f . 

4. cos A — l/f. 



7. cot A = %. 

8. cot A = §. 



11. sin A = 



2n 



1 + rc 2 



12. cos A — \/l 



FUNCTIONS OF SPHERICAL ANGLES. 33 

Find the value of the other functions : 

13. Given sin 30° = £. 16. Given tan 45° = 1. 

14. Given sin 45° = %y'2. 17. Given sin 90° = 1. 

15. Given sec 60° = 2. 18. Given sec 45° = j/2. 
Find the other functions from the following equations : 

19. sin A == 2 cos A. 22. tan A = 4 cot A. 

20. sin A = f cos A. 23. tan A = m sin A. 

21. tan A = \ sec A. 24. sec A = n tan J.. 
Express the values of the other functions in terms 

25. Of sin .4. 27. Of tan A. 29. Of sec A. 

26. Of cos A. 28. Of cot A. 30. Of esc A. 

31. Given sin A cos A = .8, to find sin J. and cos A. 

32. Given sin J. cos A = £-i/3, to find sin A and cos A. 

23. Given sin J. (sin A — cos .4) — ^-, to find sin A and cos A. 

Trigonometrical Functions of Special Angles. 

28. We will now show how to find the sine, cosine, etc. 
of some particular angles. 

I. The sine, cosine, etc. of an angle 
of 45°. 

In the right triangle ABC, suppose 
the angle A equals 45°; then angle 
B = 45° and AC = BC. Now, 

AC* + BC 2 = AB 2 , or 2BC 2 = IW. Fig. 4. 

Hence, = i ; and — - = -— . 

AB 2 2? AB t/2 

Therefore, 

sm45° = — == _ ; andcos45° = I ^ = — 

Also, tan 45° = — = 1 ; and cot 45° = j± = 1. 




34 



PLANE TR IG ONOMETR Y. 



ATI AR 

And, sec 45° = ^ = T /2 ; and esc 45° = |^ = j/2. 



vers 45 c 



1— cos 45 ( 



1- 



1/2 



II. The sine, cosine, tangent, etc. of an angle 
of 60°. 

In the right triangle AB C,let the 
angle A = 60° ; then angle B = 30°. 
Produce 4(7 to A' making CA' — 
AC. Then ABA' is an equilateral 
triangle ; and AB = A'B = AA"y ^ 




and AC =±AA'=± AB. 
Then, cos 60° = 4£ 



£45 



45 AB 



Fig. 5. 



sin 60° = 1/1 -cos 2 60< 



i/r 



tan 60° 



cot 60° = 



sin 60° 
cos 60° 

1 

tan 60° 



1 
v3' 



sec 60° = — ~ = 1 -T- \ = 2. 

cos 60° 



esc 60° 



|/3 

2 



2 

/ o 

l/3 



sin 60° 
vers 60° = 1 - cos 60° = 1 — \ = J. 

III. 27^ sme, cosine, tangent, etc. of an angle 
of 30°. 

From Art. 26 and the previous solution, 

sin 30° = cos 60° = \ ; cos 30° = sin 60° = \y/%. 



tan 30° = cot 60 c 



fo/3 ; cot 30° = tan 60° = i/3. 



sec 30 c 



esc 60° = |t/3 ; esc 30° = sec 60° «* 2. 



THE IDEA OF PROJECTIONS. 



35 



The Idea of Projections. 

29. We introduced the trigonometrical functions as re- 
lated to a right triangle. We now present a more general 
conception of the subject by the use of projections. 

30. If a line is drawn through A, 
perpendicular to AC, and BD is drawn 
parallel to AC, then AD = BC is the 
projection of AB on PQ, and AC is the 
projection of AB on A C. Calling the 
line AD or BC the vertical projection, 
and AC the horizontal projection, we 
can define the trigonometrical functions as follows : 

1. The sine is the ratio of the vertical projection of a 
line AB to the line AB. 

2. The cosine is the ratio of the horizontal projection of a 
line AB to the line AB. 

3. The tangent is the ratio of the vertical projection of a 
line AB to its horizontal projection. 

4. The cotangent is the ratio of the horizontal projection 
of a line AB to its vertical projection. 




EXERCISES IV. 

1. Given tan A = cot A, to find angle A. 

Solution. — By Art. 26, cot £= 1 — tan A ; hence tan A = \ -r- 
tan A ; hence tan 2 J. = 1, or tan A=\. Hence, Art. 28, A = 45°. 
Also infer by Art. 24. 

Find A in the followina; : 



2. Given sin A = cos 2A. 

3. Given cos A = sin 2 A. 

4. Given tan 2i = cot 2A. 



5. Given cos A = sec A. 

6. Given sin 2 A = esc 2 A. 

7. Given sin A = cos 3A. 



36 



PLANE TRIGONOMETRY. 



SECTION III. 




FUNCTIONS OF ANGLES IN GENERAL. 

31. The definitions of sine, cosine, etc., given in Art. 23, 
apply only to acute angles. But the angles of triangles are 
often obtuse ; hence it is necessary to take a more general 
view of angular magnitude and their functions. 

32. If in the diagram we suppose OA to revolve from 
the position OA to 0A X , from right 

to left, in the direction of the arc 
A A x , it will describe a right angle, 
or an angle of 90°. When OA arrives 
at OA 2 , it will have described two 
right angles, or an angular magni- 
tude of 180° ; at A s , three right angles, 
or 270° ; at OA, four right angles, or 
360°. 

33. If the line OA should continue its revolution, when 
it arrives at OA Y again it will have described five right 
angles, or 450° ; and in this way we ma}' conceive of an 
angular magnitude of gny number of degrees. Similarly, 
we may have arcs of one or more circumferences. 

34. It thus becomes necessary to extend the meaning 
of the trigonometrical functions and determine their values 
for different positions of the line AB (see Fig. 8) ; that is, 
for angles greater than 90°. 

35. For this purpose all angles are estimated from the 
line AN (see Fig. 8), as follows : 

1. Any angle, NAB, less than 90°, is said to be in the 
first quadrant. 



FUNCTIONS OF ANGLES IN GENERAL. 



37 




2. Any angle, NABi, greater than 
90° and less than 180°, is said to be 
in the second quadrant. 

3. Any angle, NAB 2 , greater than 
r80° and less than 270°, is said to be 
in the third quadrant. 

4. Any angle, NAB 3 , greater than 
270° and less than 360°, is said to be in the fourth quadrant. 

36. Now, suppose the line AN to revolve successively 
to B, Bi, B 2 , and B 3 , forming figures like those below. 
Then generalizing the conception of sine, cosine, tangent, 
etc., as given in Art. 23, we have the following definitions 
of the trigonometrical functions for the four quadrants : 
B B, 




W M 



Fig. 9 




1. Denoting the angle NAB by A, we have, as already 
explained, 

. . BC . AC , . BC , 

sin AB' cosA= ab> tan ^ = 2c ; 

2. Denoting the angle NA x Bi by A x , we have 

. A B X C X A AC, , . B 1 C l , 

smAl== Aj^ cosA ^ab^ Wl = Za>* etc - 

3. Denoting the angle NA 2 B 2 by A 2 , we have 



/ _D 2 O2 i A. O2 

sin A 2 = —r~77\ cos A 2 = 



, A B *°* ♦ 



38 PLANE TRIGONOMETRY. 

4. Denoting the angle NAB S by A- s , we have 

. , B 3 Q , AC, . . B 3 Q . 

• sin A 3 = ~jg\ cosily =^g; tan Ai=s ~A~o'> etc * 

37. From this general conception of trigonometrical 
functions we may give the following general definitions to 
these functions : 

1. The sine of an angle is the ratio of the vertical projec- 
tion of the moving radius to the radius. 

2. The cosine of an angle is the ratio of the horizontal 
projection of the moving radius to the radius. 

3. The tangent of an angle is the ratio of the vertical pro- 
jection of the moving radius to the horizontal projection. 

4. The cotangent of an angle is the ratio of the horizontal 
projection to the vertical projection. 

5. The secant of an angle is the reciprocal of the cosine. 

6. The cosecant of an angle is the reciprocal of the sine. 

Note. — If we call the horizontal projection of the moving radius 
the abscissa, and the vertical projection the ordinate, and the line AB 
the moving radius, we shall have the following simple general defi- 
nitions of sine, cosine, etc. : 

1. The sine of an angle is the ratio of the ordinate to the radius. 

2. The cosine of an angle is the ratio of the abscissa to the radius. 

3. The tangent of an angle is the ratio of the ordinate to the 
abscissa. 

4. The cotangent of an angle is the ratio of the abscissa to the 
ordinate. 

Algebraic Signs of Trigonometrical Functions. 

38. In order to distinguish the trigonometrical functions 
of the different quadrants, it has been found convenient to 
use the signs plus and minus as we do in Algebra. The 
principles by which the signs of the functions are deter- 
mined will now be explained. 



ALGEBRAIC SIGNS OF TRIG. FUNCTIONS. 



39 



B x 


c, 




/ c 


B 




c 2 


/& 


V c, 




B 2 






• ^^^ 


\ 



Q 
Fig. 10. 



39. Suppose two lines, as MN 
and PQ, to intersect each other at 
right angles in the point A. Then, 

1. All lines estimated upward from m\— \rr ^ rf-HiV' 

MN are positive ; and all lines esti- 
mated downivard from MN are nega- 
tive. 

2. All lines estimated from the ver- 
tical line PQ toward the right are positive ; and all lines esti- 
mated from PQ toward the left are negative. 

Thus, in Fig. 10, A C and BC are plus; B^ is plus; AQ is 
minus; B 2 C 2 is minus ; and B A C 3 is minus. The sign of AB 
is not supposed to change for the different positions AB X , 
AB 2 , etc. 

40. These principles, combined with the definitions of 
Art. 37, enable us to determine the algebraic sign of the 
trigonometrical functions of angles in the four quadrants. 
Thus, in the expressions of Art. 36 : 

1. Since BC and AC are both positive in the first quad- 
rant, the sine, cosine, tangent, etc. of A are plus. 

2. Since B X C^ is positive, and A(\ is negative, in the 
second quadrant the sine is plus, the cosine minus, the 
tangent (= sine -r- cosine) is minus ; etc. 

3. Since B 2 C 2 and AC 2 are both negative in the third 
quadrant, the sine and cosine are both minus, the tangent 
(= sine -r- cosine) is plus ; etc. 

4. Since B 3 C 3 is negative and AC 3 is positive in the fourth 
quadrant, the sine is minus, the cosine plus, the tangent 
minus; etc. 

41. If we let A, A x , A 2 , and A s denote respectively the 
angles CAB, CAB h etc., as in Art. 36, we have the following : 



40 



PLANE TRIGONOMETRY. 



1. Sin A is -f ; cos ^4 is + ; tan ^4 is -f ; cot A is -f ; 
sec ^4 is + ; etc. 

2. Sin A x is -f ; cos ^ is — ; tan A x is — ; cot A x is — ; 
sec A x is — ; etc. 

3. Sin A 2 is — ; cos A 2 is — ; tan A % is -f ; cot ^4 2 is -f- ; 
sec A 2 is — ; etc. 

4. Sin .4 3 is — ; cos ^4 3 is 4- ) tan ^4 3 is — ; cot A z is — ; 
sec A 3 is + ; etc. 

42. These values of the trigonometrical functions in the 
different quadrants are concisely represented in the accom- 
panying table : 





I. 


II. 


III. 


IV. 




Sine and Cosecant ....... 


-t- 


- 


- 


Cosine and Secant ....... 


+ 


- 


+ 


Tangent and Cotangent .... 


+ 


_ 


+ 


- 



43. The following exercises will serve to fix these prin- 
ciples clearly in the mind. Let the student illustrate them 
with a diagram. 

EXERCISES V. 

Show the sign of each function of an angle, 

1. Of 60°. . 3. Of 100°. 5. Of 200°. 7. Of 300°. 

2. Of 75°. 4. Of 150°. 6. Of 250°. 8. Of 320°. 

9. In what quadrant is 127°? 256°? 295°? 470°? 510°? |tt? 
fir? 3tt? tt + 40 ? tt + 100? 2tt4-60 o ? mr-f-45? 

10. Give the quadrant of angle A, if sin A is -}-> and tan A is — . 

11. Give the quadrant of angle A, if cos A is — , and cot A is -f-. 

12. Give the limits of angle A, if tan A is — , and esc A is +. 

13. Give the limits of angle A, if tan A is -{-, and sec A is — . 



FUNCTIONS OF NEGATIVE ANGLES.. 41 

14. If tan A = — f , and cos A is negative, tell the quadrant of A, and 
find values of all the functions of A. 

15. If sin A= — f, and tan A is positive, tell the limits of A, 
and find the values of all the functions of A. 

16. In a triangle, which functions may be negative, and when? 

17. In a triangle, which functions will determine the angle, and 
which will not ? 

18. For what angle in each quadrant are the absolute values of 
the sine and cosine the same? 

Functions of Negative Angles. 

44. Angles may also be regarded as positive and nega- 
tive when reckoned in opposite directions. Thus, in Fig. 7, 
if we regard angles reckoned from OA around in the direc- 
tion of AAi as positive, angles reckoned in the opposite 
direction towards AA 3 may be regarded as negative. 

45. Suppose, in Fig. 11, BCB Z is 
perpendicular to CC h and BC — B S C; 
then the sides and angles of the two 
triangles BAC and i^Care respect- 
ively equal. Let A denote angle CAB, 

Fi°\ 11. 

then — A will denote angle CAB S . 

BC B C 

1. Now, sin A = — ; and sin (— A) = -~ ; 

and i?Cand BIC are numerically equal, but have opposite 
signs (Art. 41) ; hence the results are equal with opposite 
signs. 

Therefore, sin (— A) = — sin A. 

AC AC 

2. Also, cos A = -r= ; and cos (— A) = -r~ ; 

A.±> AB 2 

and the lines are equal with the same signs. 




42 PLANE TRIGONOMETRY. 



Therefore, cos (= A) = cos A. 

o at ± , a\ sin (—.4.) —sin A . 

6. Also, tan (— A) = 7 77 = — = — tan A. 

cos (— A) cos A 

a ai x/ iN cos (—A) cos A . . 

4. Also, cot (— A) = -r— 7 7' = — - = — cot A. 

sm (— ,4) — sin ^4 

5. Also, sec (— A) — 7 7: = -. = sec A. 

J cos (—-4) cos .4 

6. Also, esc (— ^4) = - — 7 77 = — 7 = — esc A. 

' v J sm (— J.) — sm J. 



Extension of Fundamental Formulas. 

46. The Fundamental Formulas of Art. 26 were derived 
for acute angles, but it may be readily shown that they 
apply to angles of any magnitude. 

1. Thus, For. [1> Art. 26, 

• sin 2 A + cos 2 ^4 = 1, 
holds for all values of A ; for whether a and b are plus or 
minus, a 2 and b 2 are always plus, and c 2 is plus ; therefore 
the formula is always true. 

2. So also For. [2], Art. 26, 

. sin A 

tan A = 7, 

cos A 

holds for all values of A ; for both members equal a-f5 
(see Art. 26), and hence will be equal whatever be the 
signs of a and b. 

3. In the same manner all the formulas of Table I., Art. 
27, are shown to be true for any value of angle A. 

Note. — It will be interesting to the student to derive these form- 
ulas for angles terminating in each one of the four quadrants. 



REDUCTION OF FUNCTIONS, ETC. 



43 



Reduction of Functions to the First Quadrant. 

47. Haying seen that the trigonometrical functions apply 
to angles of any magnitude, we shall now show that the 
functions of all angles greater than a right angle can be 
reduced to functions of angles less than a right angle. 

48. Suppose, in Fig. 12, that the 
diameters BB 2 and B x Bz are drawn, 
making equal angles with MN. 
Then the triangles BAC, B,AC h 
B 2 AC 2 , and B 6 AC 3 are equal. De- 
note the angle NAB by A. 

Then angle NAB, = 180°- A. 



M 



By 


y' 


''" 


*~* v °> N 


B 


( 


c\ 




/ a 


\ 


\ 


c 2 


A 


\° 3 


/ 


*k 




yB 3 






*-... 


..--•* 





JV 



1. Now, sin NAB, = ~ 
AB X 



and 



Fig. 12. 

_BC 
AB 



But BC and BiC x are equal in magnitude and have the 
same sign. 

Hence, sin (180° — A) = sin A. 

2. Also, cos NAB, = —~r • and cos A = -77: : 
AB X ' vli? 

Now, ^.G and ^4C are equal in magnitude, but have 
opposite signs, 

Hence, cos (180° — A) — — cos A. 

49. The signs of the other trigonometrical functions may 
be found in a similar manner from the figure ; but a sim- 
pler method is to use the results already obtained, as given 
in Art. 26. Thus, 

sin (180° — ^4) sin A 



3. tan (180° - A) = 

4. cot (180° - A) = 



cos (180° — A) — cos A 
cos (180° - A) -cos A 
sin (180°- A) " sin .4 



= —tan A. 



cot A. 



44 



PLANE TRIGONOMETRY. 



5. sec (180° -A) = 

6. esc (180° - A) = 

7. vers (180° - A) = 



1 



cos (180° 
1 



A) 



cos A 



= — sec A. 



sin (180° — A) sin A 



1 



esc A. 

1 -f cos A. 



cos (180° — A) 

50. From Fig. 12, we see that the angle NAB 2 = 180° -f ^4, 
and angle NAB 3 = 360° — J. ; hence we can find the trig- 
onometrical functions of 180° -f A and 360° — A, as we 
found them for 180° — A in Arts. 48 and 49. These are 
given in Table II. 

51. Again, in Fig. 13, suppose 
the radii so drawn that the angles 
NAB, PAB h QAB 2 , and QAB 3 are 
all equal; then the triangles BAC, M 
B,AC U B 2 AC 2 , etc., are equal. De- 
note the angle NAB by A ; then 
reckoning from N around toward 
the left, 



B X^ 




J\ 5 


1 \c t 


\ 




I \C. 


/A 


\ ,C 1 


i%( 




^/z* 



Q 
Fig. 13. 



NAB, = 90° + A ; 



NAB, = 270° — A. 



NAB, = 270° -f A. 



1. Now, sin NAB, = 



B l C l AC 



Hence, 



AB, AB 

sin (90° + A) = cos A. 



= cos A. 



2. Also, coa NABi = 



AC, 
AB, 



BC 
AB 



sin A. 



Hence, 

3. Also, 



and 



cos (90° + A) = — sin A. 
tan (90° + A) = — cot A; 
cot (90° + A) = — tan A. 



EXTENSION OF FUNDAMENTAL FORMULAS. 45 



52. In a similar manner we may find the values of the 
functions of 270° — A, and 270° + A. All of these are em- 
braced in the following table : 



Table II. 



tan .4, 
esc A, 
sec A. 



Angle = 90°+ A. 
sin = cos A, cot = — 
cos = — sin A, sec = — 
tan = — cot A, esc = 

Angle = 180° — A. 
sin = sin A, cot = — cot A, 
cos = — cos A, sec = — sec A, 
tan = — tan A, esc = esc -4. 

Angle = 180° + A. 
sin = — sin A, cot = cot A, 
cos = — cos J., sec = — sec A, 
tan = tan A. esc = — esc A. 



Angle = 270° — A. 
sin = — cos A, cot = tan A, 
cos = — sin A, sec = — esc A, 
tan = cot A, esc == — sec A. 

Angle = 270° + A. 
sin = — cos A, cot = — tan A, 
cos = sin A, sec = esc A, 
tan = — cot A, esc = — sec A. 

Angle = 360° — ^. 
sin = — sin A, cot = — cot A, 
cos = cos J., sec = sec'^1, 
tan = — tan A, esc — — esc A. 



Note. — It will be well to have students derive all the values in the 
above table. These values can be easily remembered by observing 
that when the angle is connected with 180° or 360°, the functions in 
both columns have the same name ; but when connected with 90° or 
270°, they have different names. 

53. From what has now been presented, we see that the 
trigonometrical functions of angles of any magnitude may 
be expressed in functions of angles less than 45°. The 
same is also readily shown to be true of negative angles. 

Thus, sin 120° = sin (90° + 30°) = cos 30°. 
tan 223° = tan (180° + 43°) = tan 43°. 
cot 304° = cot (270° + 34°) = - tan 34°. 



46 PLANE TRIGONOMETRY. 

54. The functions of 360° -f %, it is readily seen, are the 
same as those of x, since the moving radius has the same 
position in both cases. In general, if n denotes any pos- 
itive whole number, 

The functions of {n x 360° -f- x) are the same as those of x. 

55. Hence, when the angle is greater than 360°, we may 
subtract 360° one or more times until we obtain an angle 
less than 360° ; and the trigonometrical functions of this 
remainder will be the same as that of the given angle. 
This remainder being less than 360°, its functions can be 
expressed in functions of an angle less than 45°. 

56. From the principle that the functions of all angles 
can be expressed in function of angles less than 45°, in the 
tables of sines and cosines we have only positive angles or 
arcs, and those of less than 45°. 

EXERCISES VI. 

1. Express the sine and cosine of 145° in functions of an angle less 
than 45°. 

Solution.— Sin 145° = sin (180° — 35°) = sin 35°. Also, cos 
145° = cos (180° — 35°) — — cos 35°. 

Express the following in functions of positive angles less than 45° : 

2. Sin 170°. . 9. Sec 246°. 16. Tan f tt. 

3. Cos 105°. 10. Csc 395°. 17. Cotf*-. 

4. Tan 125°. 11. Sin 412°. 18. Sin (— 60°). 

5. Cot 204°. 12. Cos 846°. 19. Cos (— 130°). 
• 6. Tan 300°. 13. Sin (— 35°). 20. Tan (—200°). 

7. Sin (tt + a). 14. Sin (2;r + a). 21. Cot (— 250°). 

8. Cos (tt -f a). 15. Cos (2 tt — a). 22. Sec (2w it +- a). 



EXTENSION OF FUNDAMENTAL FORMULAS. 47 



23. Derive the following table of values 



Angle . . 


30° 


45° 


60° 


120° 


135° 


150° 


210° 


225° 


Sine . . . 


1 
2 


1 

V2 


1/3 
2 


1/3 

2 


1 

V'2 


1 

2 


1 
2 


1 
1/2 


Cosine . . 


1/3 

2 


1 
1/2 


1 

2 


1 
2 


1 

1/2 


1/3 

2 


1/3 
2 


1 

1/2 


Tangent . 


1 
1/3 


1 


1/3 


-1/3 


— 1 


1 
1/3 


1 

1/3 


1 
1/3 


Cotangent 


1/3 


1 


1 
1/3 


1 
1/3 


— 1 


-1/3 


1/3 


1/3 


Secant . . 


2 
1/3 


1/2 


2 


— 2 


-1/2 


2 
1/3 


2 
1/3 


2 
1/3 


Cosecant . 


2 


•2 


1 


2 

1/3 


1/2 


2 


— 2 


— 2 



Extension of Formulas of Table II. 
57. The formulas of Table II. 
were derived on the supposition 
that the angle A is less than 90° ; 
but they are true whatever is the 
value of A. 

In order to prove this we will 
show first that they are true for 
90° + A, when A is obtuse. Let 
the angle NAB l be denoted by A. t 
Draw BB 2 perpendicular to B^s', then 
angle NAB 2 = 90° + A. 

B 2 C 2 AQ 




Now, sin (90° + A) = 
Hence, 



AB 2 AB X 

sin (90° + A) = cos A. 
Similarly, it may be shown that 

cos (90° + A) = — sin A. 



cos A. 



48 PLANE TRIGONOMETRY. 

These two formulas correspond with those of Table II., 
and the other formulas drawn from these will also corre- 
spond with those of the Table. 

Hence all the formulas of 90° + A, when A is obtuse, 
are the same as those when A is acute. Similarly, it can 
be shown that they are true when A terminates in the 
third or fourth quadrant. Therefore they are universally 
true. 

58. In a similar manner it may be shown that all the 
other formulas of Table II. are true for any value of the 
angle A. 

EXERCISES VII. 

1. Express sine and cosine of 257° in functions of an angle less 
than 45°. 

Solution.— Sin 257° = — cos (270° — 257°) = — cos 17°. 

Note. — The difference between this method of solution and that of 
Art. 56 will be readily seen. 

Find the functions of the following in angles less than 45° : 

2. 108°. 5. 196°. 8. 240°. 11. —125°. 

3. 136°. 6. 215°. 9. 318°. 12. —265°. 

4. ^ — 30°. 7. 7t — |tt. 10. 2tt — |tt. 13. 30° — 2tt. 
Find the functions of the following in terms of the functions of x : 

14. x _ 90°. 17. x — 360°. 20. x + 450°. 

15. x — 180°. 18. x + 360°. 21.x — 540°. 

16. x — 270°. 19. x — 450°. 22. x + 540°. 

Limiting Values of Trigonometrical Functions. 

59. The Limiting Values of trigonometrical functions are 
their values at the beginning and end of the different 
quadrants. 

These values are determined by the principle that the 



LIMITING VALUES OF FUNCTIONS. 



49 



value of a variable up to the limit is equal to its value at the 
limit. 

60. In Art. 23 we have 

sin A = - ; and cos A = - • 
c c 



Now. if A = 0, a — , and b = c. 

Hence, sin = - = : and cos = - 
c e 



1. 



61. In Art. 23 we have 

sin A 



tan A = 



cos A ' 



-, , , cos A 

and cot A = — 

sm A 



Hence, 



tan == - = ; and cot = ~ = oo . 



62. In Art. 53 we have 

sin (90° + 4) = cos A ; and cos (90° + A) = — sin 4. 
Hence, supposing ^4 = 0°, we have 

sin 90° = cos 0° = 1 ; and cos 90° = — - sin = — 0. 

63. Proceeding in a similar manner, we find the limiting 
values of all the functions as expressed in the following 
table : 

Table III. 



Arc — 


0. 


Arc = 90°. 


Arc = 180°. 


Arc = 270°. 


Arc = 360°. 


sin — 





sin = 1 


sin = 


sin = — 1 


sin = — 


cos = 


1 


cos = 


cos = — 1 


cos = — 


cos = 1 


tan = 





tan = oc 


tan = — 


tan = oc 


tan = — 


cot = 


OO 


cot-, 


cot = 00 


cot = 


cot = — oc 


sec = 


1 


sec = oo 


sec = — 1 


sec = — oo 


sec == 1 


CSC = 


oc 


CSC = 1 


esc = oo 


esc = — 1 


esc = — oc 



64. From the principles now explained we can often 
determine the angle from the trigonometrical functions 
by inspection. 

4 



50 PLANE TRIGONOMETRY. 

EXERCISES VIII. 
1 . Given sin 2 a — cos 2 a = 0, to find a. 

Solution. — Transposing, we have sin 2 a = cos 2 a; whence sin 
a = cos & 5 hence a = 45° or 225°. 
Find the angle a in the following : 

2. tan a = 1. 9. cos a = — ^. 

3. sin a = 1. 10. sec 2 a = 2. 

4. cosa = — 1. 11. esc 2 a = -§ . 

5. sin 2 a + cos 3 a = 0. 12. sin 2 a = 3 cos 2 a. 

6. tan a + cot a = 0. 13. sin a -f- cos a = 1. 

7. cot a — 2 cos a = 0, 14. sin 2 a — 2 cos a + i = 0. 

8. 3 sin 2 a -f 2 cos 2 a = 3. 15. 3 sec 4 a + 8 = 10 sec 2 a. 
Prove the following : 

16. sin A == tan A cos ^4. ~ 9 1 -f- sin A l-\- sec A . 

17. tan ^1+ cot .4 = sec .4 csc A "' ^^sA l + csc^~ 

18. tan A sin A = sec A — cos A. 23. 

19. cot A cos A — esc ^4 — sin A. 



24. 



OA sin A + cos ^4 . , . 

20. — = sm A cos A. 

sec vl 4- esc A 

25. 
, sin A + tan .4 . A± . cot .4 

^1- — r~r~; 7 =sm J.tanvl. 

cot ^ + csc J. 26. sec 2 ^+tan 2 ^l=sec 4 ^— tan 4 X 



CSC 


A 




cos 


JL. 


sin 


A 




sin 2 


A. 


csc 


A 




csc 


A 




sec 


A. 




THE SUM AND DIFFERENCE OF TWO ANGLES. 51 



SECTION IV. 

THE SUM AND DIFFERENCE OF TWO ANGLES. 

65. We shall now find formulas for the trigonometrical 
functions of the sum and difference of two angles. 

Let the angle AOB be denoted by 
A and the angle B OC by B ; then the 
angle AOC= A + B. 

On 00 take any point 0, draw 
CD _L to OA, CN _1_ to OB, MN _L 
to CD, and NE _L to OA. Then the 
angle CNM is the complement of 
MNO, or NO A; therefore angle NCM = angle A. 

1. Now, CD = NE + CM. 

Hence, OOsin {A + B) = OiV sin 4 + CNcos A Art. 23. 

= O0cosi?sin^4-1-O(7sml> cos.A 

Whence sin (A + B) = sin A cos B + cos A sin B. [9] 

2. Again, OD = OE — MN. 

Hence, 00 cos (A + B)= ON cos 4 - iVOsin A 

= 00 cos BcosA— OC sin B sin ^4. 
Whence, cos (A + B) = cos A cos B — sin A sin B. [10] 

66. These two formulas express the value of the sine 
and cosine of the sum of two angles in terms of the sines 
and cosines of the single angles. Enunciated in a theorem, 
the first gives 

The sine of the sum of two angles is equal to the sine of the 
first into the cosine of the second, plus the cosine of the first into 
the sine of the second. 



52 



PLANE TRIGONOMETRY. 




67. Again, in Fig 16, let the 
angle A OB be denoted by A, and 
the angle BOC by B; then angle 
NCM = angle ENC = A (B. I. 
Th. 15). 

3. Now, CD=NE-MC. 

Hence, 0(7 sin (A — B) = OA^sm ^4 — A^C cos A 

= 0(7cosi?sm^4 — 0(7sini?cos^. 
Whence, sin (A — B) = sin A cos B — cos A sin B. [11] 

4. Again, OD = OE + IfA". 

Hence, OCcos(4— £) = OA T cos A + A r Csin A 

= OOcos B cos ^4 -f 0(7 sin i? sin A 
Whence, cos (A — B) = cos A cos B + sin A sin B. [12] 

5. From Table I., For. 4, and formulas [9] and [10], 



tan {A -f B) 



sin {A + B) sin A cos i? + cos v4 sin i? 



cos {A + i?) cos A cos J5 — sin ^4 sin J5 

Dividing both terms of last member by cos A cos B, 
we have 



tan (A + i?) = 



sin ^4 cos i? cos ^4 sin i? 

cos A cos i? cos y4 cos B 

sin ^4 sin B 



cos ^4 cos J5 
Cancelling common factors, and reducing, we have 

tan A -f tan B 



tan (A -f- B) == 



1 — tan A tan B 



[13] 



6. Substituting — B for B in formula [13], and reducing, 

we have 

tan A — tan B 



tan ( A — B) = 



1 + tan A tan B 



[14] 



FORMULAS FOB DOUBLE AND HALF ANGLES. 53 

7. Dividing formula [10] by [9], and reducing as in [13], 
we have 

x , a , -r^ cot A cot B — 1 r _, 

cot(A+B) = cotB + cotA - C 15 3 

8. Substituting — B for B in formula [15], and reducing, 
we have 

wa t>n cot A cot B 4-1 _,... 

cot(A-B) = cotB _ cotA - [16] 

68. These eight formulas may be considered as the 
Fundamental Theorems of Trigonometry. 

Note. — For [14] can be derived like [13], and [16] like [15]. 



Formulas for Double and Half Angles. 

69. We now proceed to derive from these fundamental 
theorems the trigonometrical formulas for double and half 
angles. 

1. Making A = B'm formulas [9], [10], [13], and [15], 
we have 

sin 2 A = 2 sin A cos A [17] 

cos 2 A = cos 2 A — sin' 2 A [18] 

, A 2 tan A -, _- . nA cot 2 A — 1 rn ._ 

tan2A= T3btfiP»l 0Ot2A = 17otT ™ 

2. If in [18] we put 1 — sin 2 A for cos 2 A, and 1 — cos 2 A 
for sin 2 A, we have 

cos 2 A = 1 — 2 sin 2 A. cos 2 A = 2 cos 2 4 — 1. 

Whence, 



• a ^ 1 — cos 2 4 ro .-, . /I 4- cos 2 A _ ' 
an A = \ [21] cos A = ^/— ~ ^ [22] 



54 PLANE TRIGONOMETRY. 

Dividing [21] by [22], and then [22] by [21], multiplying 
numerator and denominator by the denominator, and re- 
ducing, we have 

4. a siri 2 A rocn sin 2 A ro ., 

tanA= l-fcos2A ^ COtA== r^o-s-2A^4] 

3. Substituting \A for J. in [21], [22], [23], and [24], 
we have 



sin 



VI — cos A r _ r -. . . /I + cos A rnn -. 
[25] cosiA-^-^ [26] 



Taking reciprocals of [27] and [28], we have 

, . . 1 4- cos A r _ nn , , A 1 — cos A _ - 

cot A A = — ^ — r — - [29] tan U= — : — -. — [30] 

2 sin A L J - sin A L J 



70. Sums and Differences of Functions. 

1. Adding and subtracting formulas [9] and [11], and 
doing the same with [10] and [12], we have 

sin (A + B) + sin (A — B) = 2 sin A cos B, (1) 

sin (A + B) — sin (A~-B) = 2 cos .4 sin B, (2) 

cos (.4 + B) + cos (J. — B) = 2 cos ^ cos B, (3) 

cos (4 — B) — cos (4 + E) = 2 sin yl sin J5. (4) 

2. Now, making 

A 4- 5 — p and ^ — B = #, 
whence, .4 = \ ( j) 4- <?) and B = ^(p — q); 

and substituting these in the above, and we have, 

sin p 4- sin q — 2 sin \ (p 4- (?) cos \ (p — q), [31] 
sin p — sin q = 2 cos J Cp 4- #) sin | (_/> — g), [32] 
cosjp 4- cos q — 2 cos ^ (j? + q) cos ^ (j? — q), [33] 
cos g — cosj9 = 2 sin \ (p 4- g) sin J Q) — g). [34] 



FORMULAS FOR TWO ANGLES GENERALIZED. 55 

3. Now dividing [31] by [32], 

sin ff+sin g ^ sin \ (p+q) cos -j- (p— g) _ tan \(p -f g) _ .. 
sin_p — sing cos|(j?+g)sin |(p — g) tan|-(^ — g) L J 
In a similar manner, we obtain 

sin j?+sin g _ 2 sin j- (j?+g) cos | Q— g) _ tan i ( „ + ,) [36] 
cos_p-f-cosg 2cos|-(j?+g) cos-^-Q) — g) 2 L 

sinp—sin g = 2 sin | (p— a) cosj (l> + g) ==tan j_ f <> rgy-, 
cosj>-fcosg 2cos-^(j?-f-g)cos J(_p — g) 2 L J 

sin p+sin g ^ 2 sin i (ff+g) cos \ (p— g) _ cos j- (j?— g) |-g g -, 
sin(_p+g) 2sini(p+g)cos| (jp+g) cos| (p+q)' L ° J 



sinj) — sing 2sin^-(j) — g) cos ^ (p+q) $m^(p— g) 



[39] 



sin(j?+g) 2 sin ^-(jp+g) cos j- Cp+g) sin^-(^+g)' 

[40] 



sin (_p — g) 2 sin | ( p — g) cos j- (p — g) cos \ (p — g) 



sin p — sin g 2 sin | (p — g) cos i (p+q) cos^(p+ qY 

71. These formulas may be enunciated in propositions ; 
thus formula [35] gives 

The sum of the sines of two arcs is to the difference of their 
sines as the tangent of one-half of the sum of the arcs is to the 
tangent of one-half of their difference. 

Formulas for Two Angles Generalized. 

72. In the demonstration of Arts. 65 and 67 both A 
and B, and also their sum, are assumed to be acute angles. 
These formulas, however, are entirely general, as may 
be readily seen. 

1. If the sum A -f B is obtuse, 
A and B being acute, as in Fig. 17, 
the proof is the same as in Art 65, 
except that the sign of OD will be 
negative, as NM is greater than 
OE. The formulas for sin (A -f B) and cos (A -f E) are 




56 



PLANE TRIGONOMETRY. 




therefore true for all acute angles. Formulas [11] and [12] 
may be readily derived from formulas [9] and [10] by sub- 
stituting — B for B ; hence these formulas are also true 
for all acute angles. 

2. Again, let AOB = A and BOC 
= B be both obtuse angles. Draw 
CN J_ to BC produced, EN _L to 
OA, CD _L to OA', and MN parallel 
to AA'. Then 

CD = NE + CM-, whence, Art, 36, 

00 sin (y4-f 5 — 180°) 

= OAT sin (180° - A) + (Wcos (180° — A) 
= 00 cos (180°— B) sin (180° - A) 
+ OCsin(180° — 5)cos(180° — ^4). 

Whence, sin {A + B) = sin ^4 cos J5 + cos ^4 sin B. 

3. Again, considering the algebraic signs, we have 
0D= OE—MN; whence 

00 cos (.4 + B — 180°) 

= ON cos (180° — A) — iVO sin (180° — A) 
= 00 cos (180° — B) cos (180°- y4) 
— OC sin (180° — 5) sin (180° — A). 

Whence, cos (A + B) =■ cos A cos B — sin A sin B. 
Substituting — B for B in each of the above formulas, 

wc obtain sin {A — B) and cos {A — B) 7 as in Art. 07. 

Hence, in the formulas of Arts. 65 and 67, and in all the 

formulas derived from them, A and B may be either acute 

or obtuse. 

Note. — Another method of proving the universality of these form- 
ulas is given in the Supplement, Art. 149. 



OTHER FORMULAS. 57 

EXERCISES IX. 

1. Given sin A = \ ; find sin \ A 5 find cos \ A. 

2. Given cos A = \ ; find cos 2 ^4 ; find tan 2 J.. 

3. Given tan \ A = 1 ; find sin vl ; find cos A. 

4. Given cot \ A — \/2 ; find sin ^4 ; find cos X 

15. Find the trigonometrical function of an angle of 15°. 

Solution.— Sin 15° = sin (45° — 30°) == sin 45° cos 30° — cos 45° 
sin 30°. Substituting the values of sin 45°, cos 30°, etc., as given in 
Art. 56, and reducing, we have an expression for sin 15°. Similarly, 
we find all the values given below. 

9. cot 15° = 2 + l/3. 

2i/2 

10. sec 15° =* •- ¥ ■ , . 
1/3 + 1 

9-1/9 

11. esc 15°^ T « 
l/3-l 

• Find sine, cosine, tangent, and cotangent of 

12.75°. 16. 90° + A. 20. 360° — A. 

13.105°. 17. 180° + A. 21.^ — 180°. 

14. 195°. 18. 18°— A. 22. 450° + A. 

15. 240°. 19. 27° + A. 23. 30° — A. 

Other Formulas. 
73. The student may now exercise his skill in demon- 
strating the following formulas. The Greek letter (thcta) 
is used by many writers to denote any angle. 

EXERCISES X. 

Prove the following: 

1. Sin (30° + 8) + sin (30° — 0) = cos 6. 



6. 


sin 15° 


_ V 3 — 1 

2 1/2 


7. 


cos 15°; 


_ V 3 + 1 
2i/2- 


8. 


tan 15° 


= 2~i/3. 



58 PLANE TRIGONOMETRY. 

2. cos (60° + 0) + cos (60° — 0) = cos 0. 

3. sin (60° + 0) — sin (60°— 0) = sin 0. 

4. sin 31° + sin 29° = cos 1°. 

5. sin 62° — sin 58° = sin 2°. 

6. tan (45° + 0) — tan (45° — 0) = 2 tan 0. 

7. tan 4- cot = 2 esc 0. 

8. esc — cot = tan ^ 0. 

9. esc 4- cot = cot i 0. 
10. cot£0 — tan | = 2 cot 0. 

1 4- tan 



11. tan (0 4- 45°) 

12. tan (0 — 45°) 



1 — tan 

tan — 1 
tan 4- 1 



-. cot — tan o a ' 

13. = cos 2 0. 

cot 4~ tan 

14. sin 3 = 3 sin — 4 sin 3 0. 

15. cos 3 = 4 cos 3 — 3 cos 0. 

16. COS (f 7T 4" 0) 4- COS (f 7T — 0) = — cos 0. 

17. cos 55° 4- cos 65° + cos 175° = 0. 

- 18. sin (n 4- 1) a -\- sin (w — 1) a — 2 sin na cos a. 

19. If A 4- 5 + O == 180°, prove tan J. -f tan i? 4- tan -C == cot 
^t cot 5 cot C. 

20. If A 4- JB 4- C = 90°, prove cot A 4- cot #4- cot C = cot .4 
cot 5 cot C. 

Note.— In 19th, tan (A + B) = tan (180° - (7) ; develop and simplify. 
Similarly, in 20th. 



THE THEOREMS OF TRIGONOMETRY. 



59 



SECTION V. 



THE THEOREMS OF TRIGONOMETRY. 

74. The Theorems of Trigonometry express the rela- 
tion between the sides and trigonometrical functions of the 
angles of a triangle. 

75. These theorems are designed for the solution of tri- 
angles. By the solution of a triangle is meant the rinding 
of the unknown parts from certain known parts. 

Theorem I. 

In any plane right triangle each side is equal to 
the product of the hypotenuse into the sine of the 
opposite angle. 

Let ABC be a right triangle, right 
angled at C; then (Art. 23), we have 



sin A = 



and 



sin B = -. 
c 




Hence, a = c sin A ; and b = c sin B. 

Cor. — In a plane right triangle each lg ' 19, 

side is equal to the product of the hypotenuse into the cosine 
of tthe adjacent angle. 

Theorem II. 

In any plane right triangle each side is equal to 
the product of the tangent of the opposite angle 
into the other side. 

In the triangle ABC we have (Art. 23), 

tan A = t ; and tan B = -. 
b' a 

Hence, a = b tan A ; and b = a tan B. 



60 



PL AXE TRIGONOMETRY. 



Cor. — In a plane right triangle each side is equal to the 
product of the cotangent of the adjacent angle into the other 




BC sin B. 



Note. — These two theorems enable us to solve the different cases 
of right triangles. 

Theorem III. 

In any plane triangle the sides are proportional 
to the sines of the opposite angles. 

Let ABC be a plane triangle 
whose angles are A, B, and (7, and 
sides opposite these angles a, b, 
and c. 

From C draw CD perpendicular 
to AB. Then in the right triangle 
ADC we have (Art. 23), 

CD ^ AC sin A and CD 
Hence, A C sin A = BC sin B, 

and AC : BC= sin B : sin A. 

In a similar manner it may be shown that 
AC: AB = sin B : sin (7, 
BC\ AB= sin A : sin C. 
If the angle B is obtuse, as in Fig. 
21, we have ^ 

CD = AC$mA, Fi s- 2L 

And CD = BC sin (180° — B) = BC sin B. 

Hence, JO sin A = BC sin B. 

Scholium. — This theorem enables us to solve a triangle 
when we have two angles and one side, or two sides and 
one angle not included by the sides. 




THE THEOREMS OF TRIGONOMETRY. 



61 



Theorem IV. 

In any plane triangle the sum of any two sides 
is to their difference as the tangent of half the 
sum of the opposite angles is to the tangent of half 
their difference. 

Let ABC be any plane triangle 
whose angles are A., B, and C, 
and sides opposite these angles a, 
b, and c. Then, Th. III., 

a : o = sin A : sin B. Fi s- 22 - 

Whence, 

a + b : a — b = sin A + sin B : sin A — sin B. 
sin A + sin B tan 4- ( A 4- B) 




But, 



[35] 



sin J[ — sin B tan |(^4 — B) 
Hence, a-\-b :a — b — tan %(A + B) : tan -J (^4 — 5). 

Scholium. — This theorem enables us to solve a triangle 
when we have two sides and the included angle. 

Theorem V. 

In any plane triangle, if a line is drawn from the 
vertical angle perpendicular to the base, then the 
whole base will be to the sum of the other two sides 
as the difference of those sides is to the difference of 
the segments of the base. 

Let ABC be any plane triangle, 
and CD a line drawn perpendicular 
to the base. 

Then Th. XL, Book IV., 

~AC 2 = AD 2 + 'DC 2 , 
and ~BC 2 = BD 2 -f DC 2 . 
Subtracting, AC 2 — BC 2 = AD 2 — BD 2 . 




62 



PLANE TRIGONOMETRY. 



Hence (B. IV., Th. X., C), 

(AC + BCXAC—BC) = (AD + BD)(AD — BD). 
Whence, AD + DB : AC + BC = AC— BC: AD — DB. 

Scholium. — This theorem enables us to solve a triangle 
when the three sides are given. 



Theorem VI. 

In any plane triangle the square of any side is 
equal to the sum of the squares of the other two 
sides, diminished by twice the product of the two 
sides and the cosine of the included angle. 

Let ABC be any plane triangle, 
and CD a line perpendicular to the 
base. 

Then Th. IV., Book 13., 

AC 2 = AB 2 + BC 2 — 2AB x BD; 
Also, BD = a cos B (Th. I. C). 
Hence, b 2 = c 2 + a 2 — 2 ac cos B. 

Cor.— From Th. VI., 

a 2 + c 2 — b 2 




Fig. 24. 



COS B = 



Similarly, cos A 



2ac 

b 2 + c 2 —a 2 
2bc 



cos C- 



a 2 + b' 



2ab 



Scholium. — These formulas can also be used to find the 
angles of a triangle when the three sides are given. 

76. The formulas of Theorem VI. may be put in a more 
convenient form. 

b 2 + c 2 — a 2 



Now, 



cos A = 



2bc 



Th. VI. 



NUMERICAL VALUE OF SINES, TANGENTS, ETC. 63 

Also, 1 — cos A == 2 sin 2 \ A. [25] 

Whence, 2 sin 2 1 ,4 = 1 — 



2bc 

(a + b — c)(a — b + c) 
~~ 26c 

Let 2s = a + 6 + c; 

Then, a + & — c = 2(s — c) and a — 6 + c = 2(s — 6). 

(s- b)(s-c) 
be 



Whence, 



sin 2 \ A = 



Similarly, cos 2 ^ A = r 



Hence 



tan 2 1 A 



(s-b)(s-c) 



s(s — a) 
By changing the letters we have 



[41] 



. .>,,■» ( s — a )( s — c ) • nn ( S — a )( S — b ) 

sin 2 4B = - J \ sin 2 |-C = - — 

2 ac 5 z 



COS 2 i£: 



<B-b) 
ac 



cos 2 iC = 



j. 2 1-r. ( S a )( S °) J. Jin 

tan 2 |B = - — ; yv , r — -; tan'^C 

2 s(s — b) ' 2 



ab 

s(s — c) 
ab 

(s— a)(s— b) 
s(s— c) 



^[42] 



SECTION VI. 

NUMERICAL VALUE OF SINES, TANGENTS, ETC. 

77. The theorems now presented show the relation be- 
tween the sides of a triangle and the trigonometrical func- 
tions of its angles. These sides are expressed in numbers ; 
hence, to solve a triangle we must find the numerical value 
of these trigonometrical functions for any given angle. 

78. When the radius of the circle is unity, the sine of 
the angle NAB (Fig. 8) is e6[ual to the straight line. I?C. 



64 PLANE TRIGONOMETRY. 

When the angle is very small, the line BC is very nearly 
equal to the arc AB ; hence the sine of a very small angle 
is very nearly equal to the arc which measures the angle. 

By dividing n = 3.1415926 by the number of minutes in 
180°, we find the length of an arc of V to be 0.0002908882. 
This arc is so small that it does not differ materially from 
the sine of the angle of which it is the measure ; hence, we 
may assume 

sin 1' = 0.0002908882. 

We then find the cosine of 1' by For. [3], Table I. 



Thus, cos V = VI — sin 2 V = .9999999577, etc. 

79. To find the sine of the other arcs, we take the for- 
mula under Art. 70, putting it in the form 

sin (a + V) = 2 sin a cos b — sin (a — &). 
Now, make b = 1', and then in succession, a equal to 1', 2' ? 
3', etc. and we have 

sin 2' = 2 sin 1' cos 1' — sin = 0.0005817764. 
sin 3' = 2 sin 2' cos V — sin V = 0.0008726646. 
sin 4' = 2 sin 3' cos 1' — sin 2' = 0.0011635526. 
sin 5' = etc. 
Substituting in a similar manner in the formula 
cos (a -f b) = 2 cos a cos & — cos (a — &). 
We find 

cos 2' = 2 cos 1' cos 1' — cos 0' = 0.9999998308. 
cos 3' = 2 cos 1' cos 2' — cos 1' = 0.9999996193. 
cos 4' = 2 cos V cos 3' — cos 2' = 0.9999993232. 
etc. etc. 

80. We may thus obtain the sines and cosines of angles 
of a.ny number of degrees arid minutes up to 45°. Then, 



NUMERICAL VALUE OF SINES, TANGENTS, ETC. 65 

since the sine and cosine of an angle are equal respectively 
to the cosine and sine of its complement, the sines and 
cosines of angles between 45° and 90° are immediately 
derived from those between 0° and 45°. 

81. The tangents and cotangents may be found from the 
sines and cosines by the formulas, 

sin a , cos a 

tan a = ; and cot a = ; 

cos a sin a 

and the secants and cosecants by the formulas, 
sec a = : and esc a = 



cos a ' sin a 

82. These numerical values of the sines, cosines, tan- 
gents, etc. of angles from 0° to 45°, arranged in a table, consti- 
tute what is called a Table of Natural Sines, Cosines, etc. 

Notes. — 1. In actual practice it is not necessary to continue the 
process of computation beyond 30° ; for by Art. 70 we . have, 
reducing, 

sin (30° + a) = cos a — sin (30° — a), 

cos (30° + a) = cos (30° — a) — sin a ; 

so that the table may be continued above 30° by simply subtracting 
the sines and cosines under 30° previously found. 

2. The values of the sines, cosines, etc. thus computed are very 
nearly but not absolutely correct. The equation, arc a = sin a = 
tan a, is true for the natural functions of 30° as far as six decimal 
places, and for 1° as far as five decimal places. For any arc a it 
has been shown that sin a lies between a and a — ^a 3 ; the values 
found above for large angles must therefore be corrected. 

3. The results can be verified and corrected by means of independ- 
ent calculations. Thus, cos 45° = j/-|-, Art 42 ; from which, by For. 
22 and 23, we can find sine and cosine of 22° 3CK, 11° 15', etc. So 
also from cos 30° =^-|/3, we can find sine and cosine of 15°, 7° 3CK, 
3° 45 r , etc. 

5 



66 PLANE TRIGONOMETRY. 

S3. By means of these natural signs the sides and angles 
of triangles can be readily determined. Thus, suppose in 
the triangle ABC, page 60, we have given a = 100 ft., an- 
gle A = 45°, and angle C = 60°, to find b. 

By Th. III., sin A : sin B = a : b. 

, Tn , a sin B 

Whence, b = — — — r- 

srni 

Now, a = 100 ; sin 5 = sin 60° = J j/3 ; sin 4 = sin 45°= 1- 
Hence, b = 100 x J ^3 = 50 1/ 3. 

84. In this example the numbers are small and the cal- 
culation easily made. In general, however, the sines, co- 
sines, etc. are expressed in large decimals, and the calcula- 
tion is exceedingly tedious. To avoid this labor, log- 
arithmic sines, cosines, etc. are used, which we shall now 
explain. 

Logarithmic Sines, Cosines, Tangents, etc. 

85. A Logarithmic Sine, Cosine, Tangent or Cotan- 
gent is the logarithm of the natural sine, cosine, tangent 
or cotangent. 

86. The logarithmic sine, cosine, etc. of an angle is 
readily computed from the natural sine, cosine, etc., as 
follows : 

1. We first find the logarithm of the natural sine or cosine. Then, 
since the sines and cosines of angles are less than unity, their log- 
arithms would have negative characteristics. In order to avoid these 
negative quantities, it has been found convenient to increase the 
logarithm by 10, so we make the characteristic 9 instead of — 1,8 
instead of — 2, etc. 

% The tangents of angles under 45° are also less than unity, and 



LOGARITHMIC TABLES AND THEIR USE. 07 

the characteristics of logarithmic tangents are also increased by 10. 
The same principle applies to logarithmic cotangents, secants, etc. 

ST. In using these logarithmic functions, therefore, we 
have the rule that for each logarithmic function added in 
forming a sum, ice must deduct 10 from that sum. 

SS. The logarithmic tangent and cotangent are readily 
derived from the logarithmic sine and cosine by subtract- 
ing the one from the other. 

Thus, tan A = -. • 

cos A 

Hence, log tan A = log sin A — log cos A. 

Similarly, log cot A = log cos A — log sin A. 

EXERCISES XI. 

1. Given sin 36° 24' = .59342, find log sin. Am. 9.773361. 

2. Given cos 64° 30 / = .43051, find log cos. Ans. 9.633984. 

3. Given log cos 65° 24' = 9.619386, find cosine. Arts. .41628. 

4. Given log tan 59° 44' = 10.233905, find tangent. Ans. .8639. 

5. Find log cos 36° 24 / from Ex. 1. Ans. 9.905739. 

6. Find log tan 36° 24' from Ex. 1 and 5. Ans. 9.867622. 

Logarithmic Tables and their Use. 

89. A Table of Logarithmic Sines, etc. is a table con- 
taining the logarithmic sine, cosine, tangent and cotangent 
of angles, increased by 10. (See Appendix, p. 17.) 

90. In the Table the degrees are given at the top and 
bottom of the page, and the minutes at the sides, in the 
column headed M. 

91. The column headed D contains the increase or de- 
crease^ for 1 second. This difference is found by subtract- 
ing the logarithmic sine, cosine, etc. of any angle from that 



68 PLANE TRIGONOMETRY. 

of the angle next exceeding it by 1 minute, and dividing 
the result by 60. 

Note. — This use of the difference is based on the principle of pro- 
portional parts, which though not rigidly correct is nearly enough 
so for practical purposes. 

92. We shall now explain the method of using the 
Tables of Logarithmic Functions. 

93. To find the logarithmic sines, cosines, etc. of angles 
or arcs. 

1 . When the angle is expressed in degrees, or in degrees and 
minutes. If the angle is less than 45°, look for the degrees at the 
top of the page, and for the minutes in the left-hand column ; then, 
opposite to the minutes, on the same horizontal line, in the columns 
headed Sine, will be found the logarithmic sine ; in that headed 
Cosine will be found the logarithmic cosine, etc. Thus, 

log sin 23° 35' 9.602150 

log tan 23° 35' 9.640027 

If the angle exceeds 45°, look for the degrees at the bottom of the 
page, and for the minutes in the right-hand column ; then, opposite 
to the minutes, in the same horizontal line, in the column marked at 
the bottom Sine, will be found the logarithmic sine, etc. Thus, • 

log cos 65° 24' 9.619386 

log tan 65° 24' 10.339290 

2. When the angle contains seconds. — Find the logarithmic sine, 
etc. as before ; then multiply the corresponding number found in 
column D by the number of seconds, and add the product to the pre- 
ceding logarithm for the sines or tangents, and subtract it for cosines 
or cotangents. 

We subtract for cosine and cotangent, because the greater the an- 
gle the less the cosine or cotangent. In multiplying the tabular 
difference by the number of seconds, we observe the same rule for 
the decimal point as in logarithms. If the angle is greater than 90°, 
we find the sine, cosine, etc. of its supplement. 



LOGARITHMIC TABLES AND THEIR USE. 



69 



2.85 
42 
119.70 to be added, 



EXERCISES XII. 

1. Find the logarithmic sine of 36° 24' 42^ 

Solution. 
log sin 36° 24', 
Tabular difference, 
No. of seconds, 
Product, 
log sin 36° 24 / 42", 

2. Find the logarithmic cosine of 64° 30' 30' 

Solution. 
log cos 64° 30', 

Tabular difference, 

No. of seconds, 

Product 

log cos 64° 30 / 30", 

3. Find the logarithmic tangent of 120° 15' 24 / 

Solution. 

180° 00' 00" 

120° 15 / 24" 

59° 44' 36" 



9.773361 



120 

9.773481 



9.633984 



4.41 

30 

132.30 to be subtract ed, 132 
9.633852 



The given angle, 

Supplement, 

log tan 59° 44', 

Tabular difference, 

No. of seconds, 

Product, 

log tan 120° 15' 24' 



4.84 
36 
174.24, to be added 



10.233905 



174 



Find the log sine of 40° 40 / 40". 
Find the log cos of 140° 30' 20". 
Find the log tan of 85° 25 / 45". 
Find the log cot of 144° 44 / 28". 



10.234079 

Ans. 9.814117. 

Ans. 9.887441. 
Ans. 11.097200. 
Ans. 10.150603. 

94. To find the angle corresponding to any logarithmic sine, 
cosine, tangent, or cotangent. 

1. Look in the proper column of the table for the given logarithm ; 
if found there, and the name of the function is at the head of the 
column, take the degrees at the top, and the minutes on the left; but 



70 PLANE TRIGONOMETRY. 

if the name of the function is at the foot of the column, take the de- 
grees at the bottom, and the minutes on the right. 

2. If the given logarithm is not exactly given in the table, then 
take the next less logarithm, subtract it from the given logarithm, 
and divide the remainder by the corresponding tabular difference ; 
the quotient will be seconds, which must be added to the degrees 
and minutes corresponding to the logarithm taken from the table, 
for sines and tangents, and. subtracted for cosines and cotangents. 

EXERCISES XIII. 

1. Find the angle whose logarithmic sine is 9.617033. 

Solution. 
Given log sine, 9.617033 

Next less in table, ' 9.616894 

Tabular difference, 4.63) 139.00(30, to be added. 

Hence the angle is 24° 27 / 30". 

2. Find the angle whose logarithmic cosine is 9.704682. 

Solution. 
Given log cosine, 9.704682 

Next less in table, 9.704610 

Tabular difference, 3.58) 72.00(20, to be subtracted. 

Hence, the angle is 59° 33' 40". 

3. Find the angle whose log sine is 9.438672. Ans. 15° 56 / 14". 

4. Whose log cosine is 9.634520. Ans. 64° 27 / 47". 

5. Whose log tangent is 10.753246. Ans. 79° 59' 24". 

6. Whose log cotangent is 11.449852. Ans. 2° V 40". • 
95. The secants and cosecants are omitted in the table, 

since they are easily derived from the sines and cosines. 
Thus, by Art. 26, 

sec A = 7, and esc A 



cos A 1 sin A 

Whence, sec A cos A==l; and esc A sin A — 1. 

Taking the logarithm and observing to add 10 to each 
logarithm, we have 

log sec A = 20 — log cos A. 
log esc A = 20 — log sin A. 



LOGARITHMIC TABLES AND THEIR USE. 71 

Hence, the logarithmic secant is found by subtracting the log- 
arithmic cosine from 20, and the logarithmic cosecant is found 
by subtracting the logarithmic sine from 20. 

EXERCISES XIV. 

1. Find the log esc of 24° 27' 34". Arts. 10.382949. 

2. Find the log sec of 54° 12 / 40". Ans. 10.232992. 

3. Prove that the log cot of an angle equals 20 minus the log tan 
of the angle, and conversely. 

Notes. — 1. The sine of an angle near 90° varies much more slowly than 
the sine of an angle near 0°, while the opposite is true of their co^nes. 
Hence, in finding an angle near 90° it is better to avoid the use of its 
sine, and in finding an angle near 0° it is better to avoid the use of its 
cosine. The tangent varies with the arc more rapidly than either its sine 
or cosine, and may be used equally well with any angle. 

2. The Tables of Logarithmic sines, cosines, etc. extend to six decimal 
places. They can be easily changed in use to five-place logarithms by 
omitting the sixth decimal and adding one to the fifth decimal when the 
figure omitted is greater than 5. Thus, for log tan 23° 35' = 9.640027, we 
may write log tan 23° 35' = 9.64003. In a similar way six-place logarithms 
may be reduced to four-place and three-place logarithms. 

Some mathematicians prefer five-place tables, and for work not requir- 
ing great accuracy even four-place and three-place tables are used. 



72 PLANE TRIGONOMETRY. 

SECTION VII. 

THE SOLUTION OF TRIANGLES. 

96. The Solution of Triangles is the process of rinding 
the unknown parts when a sufficient number of the parts 
are given. 

97. There are six parts in a plane triangle, and three of 
these, one of the three being a side, must be given to find 
the other parts. 

98. If the angles alone were given, it is clear that the 
sides could not be determined, since there could be an 
indefinite number of triangles having their angles respect- 
ively equal. 

Solution of Plane Right Triangles. 

99. In the solution of right triangles we have the four 
following cases: 

1 When the hypotenuse and one acute angle are given. 

2. When the hypotenuse and a leg are given. 

3. When one leg and either acute angle are given. 

4. When the two legs are given. 

Case I. 

100. Given the hypotenuse c and one acute angle 
A, to find the other parts. 

Method.— Let ABC denote the tri- 
angle. Then, to find a, we have, Th. I., 

. a 

sin A = - • 

c 

Whence, log a = log c + log sin A. Fig. 25. 




SOLUTION OF PLANE RIGHT TRIANGLES. 



73 



Hence, to find a, we add log c to log sin A, and find the 
number corresponding to the resulting logarithm. 
Similarly we find b ; and B — 90° — A. 

EXERCISES XV. 

1. In a right triangle ABC, given the hypotenuse c = 475, 
and angle A = 36° 34' ; find the remaining parts. 

Solution. — From the method given above we have the following 
operation : 



log c (475) = 2.676694 

log sin A (36° 340 = 9.775070 

log a = 2.451764 

a = 282.985 



log c (475) = 2.676694 
log sin B (53° 260 = 9-904804 
log 6 = 2.581498 
6 = 381.503 



Note. — In adding log sin A to log c, 10 is rejected from the sum to 
correct for the 10 which was added to the log. of the sine (Art. 87). 



37° 30'; find ^1 = 52° 30', 



14° 48', a = 



2. Given the hypotenuse c = 45.36, A = 45° 36' ; find 
a = 32.408, b = 31.736, and B = 44° 24'. 

3. Given c = 250, and B 
a = 198.338, and b = 152.19. 

4. Given c = 251.4, A = 75° 12'; find B 
243.06, and b = 64.22. 

Case II. 
101. Given the hypotenuse, c and one of the legs 
a, to find the remaining parts. 

Method.— Let ABC denote the tri- 
angle. Then, to find the angle A, 

We have sin A = - • Th. I. 

c 

Whence, log sin A = log a — log c. 
Similarly, log b = log sin 5 -f log c. 

From these A and b are readily found. And B = 
90° -A. 




74 



PLANE TRIGONOMETRY. 



EXERCISES XVI. 

1. Given the hypotenuse c = 125, and the side a = 76.095 ; 
to find the remaining parts. 

Solution. — From the method indicated above we have the follow- 
ing operations : 



log a (76.095) = 1.881357 
log c (125) = 2.096910 
log sin A = 9.784447 
A = 37° 30 / 
B = 52° 30' 



log c (125) = 2.096910 
log sin B (52° 30°) = 9.899467 
log b = 1.996377 
b = 99.169 



Note. — After subtracting it is necessary to add 10 to the result to give 
log sin A. In practice we add 10 to the minuend before subtracting. 

2. In a right triangle ABC, given c = 400, and a = 240; 
find b = 320, A = 53° 7 49", and B = 36° 52' 11". 

3. In a right triangle ABC, given c — 396, and b — 218 ; 
find A = 56° 35' 54", B = 33° 24' 6", and a = 330,59. 

4. In a right triangle ABC, given c = 126.206, and b = 
97.72; find a = 82.507, A = 40° 10' 30", and B = 49° 49' 
30". 

Case III. 

102. Given one leg, as h, and either acute angle, 
as A, to find the remaining parts. 

Method.— From Th. II., we have 

tan A = - ; 

Whence, log a = log tan A -f log b. 
Also, log c = log a — log sin A. 

Fig. 27. 




SOLUTION OF PLANE BIGHT TRIANGLES. 75 



EXERCISES XVII. 

1. In a right triangle ABC, given the side b — 200, and 
the angle A = 34° 45' ; to find the other parts. 

Solution. — From the method indicated above, we have the follow- 
ing operations : 



log tan A (34° 45') = 9.841187 
log b (200) = 2.301030 
log a = 2.142217 
a = 138.74 



log a (138.74) = 2.142217 
log sin A (34° 45') = 9.755872 
I log c = 2.386345 
c = 243.41 



2. In a right triangle ABC, given a = 364.3, A = 50° 45'; 
find & = 297.645, c = 470.433, and B = 39° 15'. 

3. In a right triangle ABC, given & = 90.5, and A = 50° 
30' ; find a = 109.78, c = 142.27, and B = 39° 30'. 

4. In a right triangle, given a = 305.34, and £ = 50° 18' 
32"; find b = 367.9, c = 478.1, and A = 39° 41' 28". 

Case IV. 
103. Given the two sides, a and b, about the right 
angle, to find the remaining parts. 

a 



Method. —We have tan A = 



V 



Th. II. 



Whence, 
Also, 



log tan A = log a — log b. 

log c = log a — log sin A, 



EXERCISES XVIII. 

1. In a right triangle, the side a= 239, side b= 188; 
find the angles and hypotenuse. 



Solution. 



log tan A = log a — log b 
log a (239) — 2.378398 
log 6 (188) = 2.2741 58 
log tan ^1 = 10.104240 
A = 51° 48' 40" 
jB = 38°11 / 20" 



log c = log a — log sin A 

log a (239) == 2.378398 
log sin A (51° 48', etc.) = 9.895409 
log c = 2.482989 
c = 304.08 



76 



PLANE TRIGONOMETRY. 



2. In a right triangle, given a = 99.98, b = 152.71 ; find 
c = 182.5, A = 33° 12', B = 56° 48'. 

3. In a right triangle, given a = 515, b = 505 ; find A = 
45° 33' 42", B = 44° 26' 18", and c = 721.28. 

4. In a right triangle, given a = 29.37, b = 37.29 ; find 
c = 47.467, 4 = 38° 13' 28", B = 51° 46' 32". 

Solution of Plane Oblique Triangles. 

104. In the solution of oblique triangles there are four 
cases, as follows : Given 

1. Two angles and a side. 

2. Two sides and an angle opposite to one of them. 

3. Two sides and the included angle. 

4. The three sides. 

Note. — In the solution, let A, B, and C denote the angles of the 
triangle, and a, b, c denote the sides opposite these angles respectively. 



Case I. 
105. Given two angles A and B and one side a, 
to find the remaining parts. 

Method.— Let ABC be the tri- 
angle. 

1. Then to find 0, subtract the 
sum of A and B from 180°. Fig" 28. 

2. To find b we have (Th. III.); 3. To find c we have 




a : b = sin A : sin B, 

Whence, b = — -r — r • 
sin ^4 



a : c = sin A : sin C, 
a sin C 



Whence, c = 



sin A 



SOLUTION OF PLANE OBLIQUE TRIANGLES. 77 



EXERCISES XIX. 

1. In the triangle ABC, given A = 32° 24', B = 40° 32', 
a = 240 ; find the remaining parts. 

Solution. — Applying the logarithms to the formulas given above, 
and substituting the numerical values, we have the following oper- 
ations : 

a= 240 log a-= 2.380211 

A = 32° 24' log sin B = 9.812840 

B = 40° 32' coloe: sin A = 0.270976 



log a = 2.380211 

log sin C = 9.980442 

£olog sin A = 0.270976 



log c = 2.631629 
c = 428.182 



A + B = 72° 56 / log b = 2.464027 

(7=107° 04' 6 = 291.09 

2. In the triangle ABC, given A = 27° 40', C= 65° 45', 
c = 625 ; find B = 86° 35', a = 318.29, & = 684.266. 

3. In A ABC, given ^ = 30° 20', 5 = 50° 10', and c = 
186.74 ; find (7= 99° 30', a = 95.62, and b = 145.39. 

4. In A ABC, given B = 51° 15' 35", (7= 37° 21', 25", 



and a = 305.296 ; find A 



91° 23', & = 238.197, c 



185.3. 



Case II. 
106. Given two sides a and b, and the angle A 
opposite to the side a, to find the remaining parts. 
Method.— In this case we proceed as follows : 
1. To find B, we have (Th. II.), 
a : b = sin A : sin B ; whence, sin B 



b sin A 



2. To find C, we have, C= 180° - (A + B). 
'3. To find e, we have, Th. III., 



a : c = sin A : sin c ; whence, c = 



a sin C 
sin A 



Discussion. — Here the angle B is determined from its sine ; and 
since the sine of an angle equals the sine of its supplement, the 



78 PLANE TRIGONOMETRY. 

angle B admits of two values, supplements of each other. We must 
therefore examine the problem to see which of the two angles (or if 
both) must be taken. 

Let ABC denote the triangle; then from c 

the principles of Geometry we have the fol- //\ 

lowing conclusions : y/ / \ 

1. If a > b, then A^> B, and B must be / /_ \ 

acute; hence there is only owe value of B, B "------'B' 

Fie 29 
and one, and only one, triangle that will satis- 
fy the given conditions. 

2. If a = b, then A = B, and* both A and B are acute, since their 
sum is less than 180°, and there is only one value of B, and only one 
triangle, and that is isosceles. 

3. If a < b, then A < B ; and A must be acute in order that the 
triangle is possible ; and if A is acute, it is evident that there are two 
triangles ABC and AB / C which satisfy the given conditions. The 
angles AB C a.ndAB'C are supplementary; hence in this case in 
finding B from sin B, we use both the angle given by the table and 
its supplement. 

4. In the formula, sin B = b sin A — a, if a = b sin A, then sin 
B = 1 ; hence B = 90°, and the required triangle is a right triangle. 

5. If a <C b sin A, then sin B > 1, which is impossible, and the 
triangle is impossible. So also if a < b and A = 90°. 

Notes. — 1. In practice the number of solutions can be usually deter- 
mined by the circumstances of the problem. When there is any doubt, 
compute the value of b sin A, and compare it with a, according to Art. 5. 

2. Or find the value of log sin B. Then, if log sin B <^ 10, there is one 
solution when a ^> b, and two solutions if a <^ b. If log sin B ^> 10, the 
triangle is impossible. 

EXERCISES XX. 

1. In the triangle ABC, given a = 75.5, b = 98.5, A = 37° 
37' ; find B, C, and c. 

Solution. — Applying logarithms to the formulas given above, and 
substituting the numerical values, we have the following operations: 



SOLUTION OF PLANE OBLIQUE TRIANGLES. 79 



a = 75.5 


b = 98.5 


A = 37° 37' 


Here a > 6 and 


log sin B < 10. 


. • . one solution. 



log 6 = 1.993436 

log sin A = 9.785597 

cqlog a = 8.122053 



log sin B = 9.901086 
£=52°46 / 48 // 
(7=89° 36' 12" 



log a = 1.877947 

log sin (7=9.999989 

colog sin A = 0.214403 

log c = 2.092339 

c = 123.69 



Note. — By constructing the triangle and examining it geometrically, 
it will be seen that there is but one solution. 

2. In the triangle ABC, given a = 150, b = 200, A = 44° 
26'; find B, C, said c. 

Solution. — Substituting in the formulas given above, we haveHhe 
following operations : 



a= 150 


log b = 2.301030 


log a = 2.176091 


2.176991 


6 = 200 


log sin A = 9.845147 


log sin C = 9.962692 9.618456 


A = 44° 26' 


colog a = 7.823909 


colog sin A = 0.154853 | 0.154853 


Here a <^b 


log sin B = 9.970086 


log c = 2.293636 


1.949400 


log sin B < 10. 


B - 68° 58' 38" 


c = 196.623 




• . two solutions. 


or 111° 01' 22" 
C= 66° 35' 22" 

or 24° 32' 38" 


or c = 89.002 





Note. — By constructing the triangle and examining it geometrically, 
ft will be seen that there are two solutions. In Fig. 29, AB'C = 68° 58' 
38", and ABC = 111° 01' 38" ; ACB' = 66° 35' 22", ACB = 24° 32' 38" ; 
AB = 89.002, AB' = 196.623. 

3. In A ABC, given a = 62.50, c = 45.96, A = 79° 21' ; 
find B = 54° 22' 22", C= 46° 16' 38", b = 51.69. 

4. In the triangle ABC, given a = 15.71, 1) = 21.12, A = 
27° 50' ; find the other parts. 

Am. B = 38° 52' 47" ; (7= 113° 17' 13" ; c = 30.906 ; 
or B = 141° 7 13"; C= 11° 2' 47" ; c = 6.447. 

5. In the triangle ABC, given a = 94.26, b =126.72, and 
A = 27° 50' ; find the values of c, B, and C. 



80 



PLANE TRIGONOMETRY. 



6. Given a = 40, b = 80, and 4 = 30° ; find the other 
parts of the triangle. 

7. Find the other parts of a triangle, given a = 80, b — 
100, and A = 60°. 



Case III. 

107. Given two sides, a and b, and the included, 
angle C, to find the remaining parts A, B, and c. 

Method.— In this case we pro- 
ceed as follows : 

1. To find A and B, we subtract 
C from 180° and divide by 2, which 
gives us the value of \ (A + B). 

We then find } (A — B) from 
Th. IV., which gives 




Fig. 30. 



tan l(A-B) = jr\ x tan iU+ $). 



Then, 
And 



i(A + B) plus $(A — E) = A. 
i (A + B) minus i (A — B~) = B. 
2. To find c, we apply Th. II., which gives 



a sin C 
sin A ' 



or c 



b sin C 
sin B 



EXERCISES XXI. 

1. In the triangle ABC, given a = 680, b =460, and C = 
84° ; find the other parts of the triangle. 

Solution. — Following the method stated above, we have the fol- 
lowing work : 



SOLUTION OF PLANE OBLIQUE TRIANGLES. 81 



a + b = 


1140 


a — b = 


220 


A+B = 


96° 


i (A + B) = 


48° 


\ {A-B)=12° 5' 


49" 


^4=60° 5' 


49" 


£=35° 54 


'11" 



log (a - b) 

colog {a + b) 

log tan i (4+.B) 


= 2.342423 
= 6.943095 
= 10.045563 


logtan£(4-.B) 

A {A - B) = 


= 9.331081 
12° 5' 49" 



log a =2.832509 

log sin C =9.997614 

colog sin A =9.062046 

log c =2.892169 

c =780.134 



2. In the triangle ABC, given a = 240, b = 360, C= 68° 
36'; find A = 39° 21' 34", B =72° 02' 26", c = 352.349. 

3. In the triangle ABC, given a = 320, b = 562, C= 128° 
04'; find A =18° 21' 21", J5 = 33° 34' 39", c = 800. 

4. In the triangle ^50, given b = 50.24, c = 43.25, A = 
40° 15'; find B= 81° 24' 25", C= 58° 20' 35", a = 32.829. 

5. If two sides of a triangle are each equal to 60 ft., and 
the included angle is 60°, what is the third side? 

6. If two sides of a triangle are each equal to 120 ft., and 
the included angle equals 120°, what is the third side? 



Case IV. 

108. Given the three sides, a, b, and c, of a plane 
triangle, to find the angles A, B. and C. 

Method.— Let fall a perpendicular upon the greater side 
from the angle opposite, dividing the triangle into two 
right triangles. Find the difference of the segments 
of the base by Theorem V. ; half this difference added to 
half the base gives the greater segment, and subtracted 
from half .the base gives the less. 

We shall then have two sides and the right angle of two 
right triangles, from which we can find the acute angles 
by Theorem I. 



82 



PLANE TRIGONOMETRY. 



EXERCISES XXII. 

1. In a triangle ABC, given AB = 60, AC — 50, and 
BC— 40, to find the angles. 

Solution. — Let ABC denote the tri- 
angle ; then AB = 60, A C = 50, B C = 
40. Then, by Th. V., ' 

AB : A C+B C=A C—B C : AD—BD, 
or, 60: 90 = 10 : AD—BB, 
Hence, 
Then, 
And 




Fig. 31. 



AB — BD = 90 X 10 ~ 60 = 15 ; 
AD = 1(60 + 15) = 37.5, 
BD = ^(60— 15) = 22.5. 



'Then, in the triangle .4 CD, to find the angle BCD, 



colog^C (50) = 8.301030 
log AD (37.5) = 1.574031 



colog^C (40) = 8.397940 
log BD (22.5) = 1.352183 
log sin BCD = 9.750123 
.-. BCD = 34° 13' 44" 



log sin ACD = 9.875061 
^(72) = 48° 35' 25" 

Hence, J = 90° — 48° 35 / 25" = 41° 24 / 35", 

J5 = 900 _ 340 13/ 44// = 550 46 / 16 // ? 

C = 48° 35' 25" + 34° 13 / 44" = 82° 49' 09". 

2. In a triangle ABC, given a = 1005, b = 1210, c = 1368 ; 
find the angles. Ans. 45° 22' 34" ; 58° 58' 19" ; 75° 39' 7". 

3. In a triangle ABC, given a = 340, & = 280, and c = 460 ; 
find the angles. 

Ans. A = 47° 23' 16" ; B = 37° 18' 31" ; (7= 95° .18' 13". 



Another Method. 
109. The angles of a plane triangle may also be found 
by means of the formulas given in Art. 76. 



SOLUTION OF PLANE OBLIQUE TRIANGLES. 83 



EXERCISES XXIII. 

1. In the triangle ABC, the side c = 800, the side b = 
600, and the side a = 400 ; required the three angles. 



Solution. — By Art. 76 we have 

Sm * A ~ be 

s = i (800 + 600 + 400) = 900. 

s — b = 900 — 600 = 300 ; 

s — c = 900 — 800 = 100. 
We then find log (s — 6), log 
(s — c), colog 6, and colog c ; their 
sum will be log sin 2 -|- J.. Divid- 
ing by 2, we have log sin \A = 
9.397940; from the Table, we find 
\ A = 14° 28' 39", whence, 
A = 28° 57' 18". 



Operation. 
log (s — b) (300) = 2.477121 
log ( s _ c ) (100) = 2.000000 
colog b (600) = 7.221849 
colog c (800) = 7.096910 
log sin 2 ^ 2)18.795880 

log sin } A = 9.397940 

| A = 14° 28 / 39" 
A = 28° 57' 18" 



The other angles may be obtained in a similar manner from the 
formulas for sin 2 \ B and sin 2 \ C. 

The other two formulas of Art. 76, which may be used in this 
case, are 



cos' 



s(s — a) 
be ' 



tan 2 \ A 



(s -i)(s-c) 
s(s — a) 



Note. — Either of these three formulas may be used ; but sin 3 £ A is less 
accurate when the half angle is near 90° ; and cos 9 £ A, when the half an- 
gle is near 0° ; while tan 3 i A is applicable for any angle. 

2. The three sides of a plane triangle are 20, 30, and 40 ; 
required the three angles. 

Ans. 28° 57' 18"; 46° 34' 03"; 104° 28' 39". 

3. In the triangle ABC, a = 200, b = 250, and c = 300; 
required the three angles. 

Ans. 41° 24' 35" : 55° 46' 16" ; 82° 49' 09". 



84 PLANE TRIGONOMETRY. 

Find the angles — 

4. Given a = 10, b = 24, c = 26. 

5. Given a = 12, & = 12, c = 12. 

6. Given a = 7, & = 8, c = 16. 

7. Given a = 2, b = |/6, c = j/3 + 1. 

Solve the following without the use of logarithms : , 

8. If b = 3, c = 2^/3, and vl = 30° ; prove (7= 90°. 

9. If a = 2]/3, & = 3 — !/3, and c = 3-|/2 ; prove 
(7= 120°. 

10. If a = 2, & = 1 + x/3, and c = -j/6 ; prove C = 60°. 

11. If a = 12, & = \ 9 ^, and A = 45° ; prove 5 = 36°. 

12. Find the angles of a triangle whose sides are in the 
ratio of 1, 2, and 3. 

Remark. — All three angles may be computed by the formulas, 
and the accuracy of the results tested by seeing whether their sum 
equals 180°. For this method the formulas for the tangent may be 
put in a more convenient form. Thus, tan 2 £ A may be written : 

(s -d)(s- b)(s — c) _ 1 f (s - a)(s - b)(s - e) \ 
s(s — a) 2 (s — a) 2 \ s ) 

If we put 

(s — a)(s — b)(s—c) „ u , . . r 

± ^ ^ J - = r 2 , we have tan I A = • 

s z s — a 

T T 

Similarly, tan -J B = , and tan \ C = • 

S s c 

In applying these formulas we may find the value of log r, and use 
it in each one of the formulas in the computation, and thus slightly 
abridge the labor of computation. 



HEIGHTS AND DISTANCES. 



85 



SECTION VIII. 



PRACTICAL APPLICATIONS. 



HEIGHTS AND DISTANCES. 

110. A Horizontal Plane is a plane which is parallel to 
the plane of the horizon. 

111. A Vertical Plane is a plane which is perpendic- 
ular to a horizontal plane. 

112. A Horizontal Line is any line in a horizontal 
plane. A vertical line is a line perpendicular to a horizon- 
tal plane. 

113. A Horizontal Angle is an angle in a horizontal 
plane. A Vertical Angle is an angle in a vertical plane. 

114. An Angle of Elevation is a vertical angle having 
one side horizontal, and the inclined 
side above the horizontal side; as 
BAD. 

115. An Angle of Depression is a 
vertical angle having one side horizon- 
tal, and the inclined side under the 
horizontal side; as CD A. 

116. Distances upon the ground are usually measured by 
a chain, called Canter's Chain. This chain is 4 rods or 66 
feet long, and consists of 100 links. Sometimes a half chain 
is used, consisting of 50 links. 

117. Angles are measured by various instruments. 
Horizontal angles are measured by an instrument called 




Fig. 32. 



86 



PLANE TRIGONOMETRY. 



The Compass. Horizontal and vertical angles are both 
measured by the Theodolite, or, what is still better for gen- 
eral use, a Transit- Theodolite. 

Case I. 

118. To determine the height of a vertical object 
standing upon a horizontal plane. 

Method. — Measure from the foot of the object any con- 
venient horizontal distance AB ; at c 
the point A take the angle of eleva- 
tion BAC; then, in the triangle ABC 
we have a side and an acute angle; 
hence, we can readily find the alti- 
tude. 

1. From the foot of a tower I meas- 
ure a horizontal line 120 feet, and at its extremity find the 
angle of elevation to be 48° 36' ; what was the height of 




Fig. 33. 



the tower? 



Ans. 136.113 feet. 



Case II. 

119. To find the distance of a vertical object whose 
height is known. 

Method.— Measure the angle of ele- 
vation to the top of the object, as 
before; we will then have a right 
triangle in which we know the per- 
pendicular and an acute angle ; hence, 
we can readily find the base. 

1. I took the angle of elevation to 
the top of a flagstaff whose height I knew to be 160 feet, 
and found it be 20° ; how far was I from the staff? 

Ans. 439.60 feet. 




Fig. 34. 



HEIGHTS AND DISTANCES. 



87 




Case III. 

120. To find the distance of an inaccessible object. 

Method. — Measure a horizontal 
base-line AB, and then take the an- 
gles formed by this line and lines 
from the object to the extremities of 
this base-line, as CAB and ABC) the 
distance AC oy BC can then be 
readily found. 

1. I am on one side of a river, and wish to know the 
distance to a tree on the other side. I measure 300 yards 
by the side of the river, and find that the two angles formed 
by this line and the lines from its extremities to the tree 
are 72° 40' and 45° 36' respectively ; required the distance 
from each extremity of the base-line to the tree. 

Arts. 243.362 yards ; 325.15 yards. 



Fig. 35. 



Case IV. 

121. To find the distance between two objects sepa- 
rated by an impassable barrier. 

Method. — Select any convenient station, as (7, and 
measure the distance from it to each 
of the objects A and B and the an- 
gle C included between these lines. 
We can then readily find the dis- 
tance AB. 

1. The distance between two trees 
cannot be directly measured : I therefore take a third posi- 
tion, from which each of the trees can be seen, and find the 
distances from it to the trees to be 300 and 250 yards re- 




88 



PLANE TRIGONOMETRY. 




spectively, and the included angle 43° 16' ; required the 
distance between the trees. Ans. 208.025 yards. 

Case V. 

122. To find the height of a vertical object stand- 
ing upon an inclined plane. 

Method.— Measure any convenient distance DC on a line 
from the foot of the object, and at 
the point D measure the angles 
of elevation; EDA and EDB, to 
foot and top of the tower. By 
means of the two triangles DEA 
and DEB we can find the height 
of AB. Fi s- 37 - 

1. Wishing to determine the height of a tower situated 
upon a hill, I measured a distance down the slope of the 
hill 400 feet, and found the angles of elevation to the foot 
of the tower 42° 28', and to the top of the tower 68° 42' ; 
required the height of the tower. Ans. 486.747. 

Case VI. 

123. To find the height of an inaccessible object 
above a horizontal plane. 

First Method.— Measure any 
convenient horizontal line AB 
directly toward the object, and 
take the angles of elevation at 
A and i>; we will then have 
conditions sufficient to find DC. Fi „ 

1. Wishing to find the alti- 
tude of a hill, I measured the angle of elevation at the 
bottom 60° 37', and 460 feet from the foot, in a right line 




HEIGHTS AND DISTANCES. 



89 



from the top of the hill and the point at the foot, and in the 
same horizontal plane as the foot, I measured the angle 
of elevation 36° 52'; required the height of the hil]» 

Ans. 597.092. 
Second Method. — If it is not 
convenient to measure a horizontal 



base-line toward the 



we may 




Fig. 39. 



measure any line AB, and also 

measure the horizontal angles 

BAD, ABD, and the angle of 

elevation ()BC. Then, by means 

of the two triangles ABD and CBD, the height CD can 

be found. 

Case VII. 

124. To find the distance between two inaccessible 
objects when points can be found from which both 
objects can be seen* 

Method. — The method of meas- 
urement is indicated in the follow- 
ing problem. The method of solu- 
tion we prefer leaving to the ingenuity 
of the pupil, that he may learn to 
think for himself. 

1. Wishing to know the horizontal 
distance between a tree and house 

on the opposite side of a river, I took the following meas- 
urement : 

.45 = 400; CAD^ 56° 30', 

BAD = 42° 24' ; ABC = 44° 36', 
• and DBC=68° 50'. 
Required the distance CD. Ans. 747.913. 




90 



PLANE TRIG ONOMETR Y. 



F-^f 




Fie. 41. 



Case VIII. 

125. To find the distance between two inaccessible 
objects when no points can be found from which 
both objects can be seen. 

Method. — The method is indicated in the following 
problem and figure. This 
case and the following one 
may be omitted with young 
students. 

1. Wishing to know the 
horizontal distance between 
two inaccessible objects 
when no point can be found 

from which both objects can be seen, two objects C andD 
are taken, 600 feet apart, from the former of which A can 
be seen, from the latter B. From C we measure the dis- 
tance CF, not in the direction DC, equal to 600 feet, and 
from D a distance DE equal to 600 feet. We then measure 
the following angles. 

CFA = 80° 16', BED = 86° 25'. 
ACF= 52° 24', BDE = 60° 24', 
ACD= 56° 36', BDC= 150° 30'. 
Required the distance AB. Ans. 1117.44 feet. 

Case IX. 

126. To find the distances from a given point to 
three objects whose distances from each other are 
known. 

Method. — The method is indicated in the problem and 
figure. 



SUPPLEMENT. 



91 



1. I wish to locate three buoys, 
A, B, and C, in a harbor, so that 
the distance between A and B is 
800 yards, between A and C 600 
yards, between B and C 400 yards, 
and from a fixed point on shore 
the angle APC shall equal 33° 45' 
and BPC 22° 30'; required the 
distances PA, PC, and PB. 

Ans. PA = 710.193; PC = 1042.522; PB = 934.291. 




Fig. 42. 



SECTION XL 

SUPPLEMENT. 

127. The Supplement presents additional matter for those who 
wish to pursue the subject further. 

Some Properties of Triangles. 
The Right Triangle. 

128. In the right triangle ABC, let b denote the base, a the alti- 
tude, c the hypotenuse, and M the area. Then, Geometry, B. IV., 

Th. 6, 

M=%ab. 

But, a — b tan A, and b = a tan B. Art. 26. 

Hence S = £ b 2 tan .4, and M=\d l tan 5. [43] 

Hence, we can find the area from A and 6 or from a and 5. From 
Ex. 1 and 2 below we can find the area having a and A or 6 and B> 
From Ex. 3 and 4 we can find the area, having given c and A or i?. 



92 PLANE TRIGONOMETRY. 



EXERCISES XXIV. 

Prove the following : 

1. M=\a 2 cot A. 3. M=lc 2 sin 21 

2. if = \ W cot £. 4. M = £ c 2 sin 2 £. 
Find the other three parts : 

5. Given B and c. 7. Given B and a. 

6. Given B and 6. 8. Given b and c. 

The Isosceles Triangle. 

129. In the isosceles triangle ABC } let h denote the altitude, a 
the equal sides, and c the base. Then we readily derive the rela- 
tions given in the following exercises : 

EXERCISES XXV. 

In an isosceles triangle find the other parts — 

1. Given a and A. 4. Given c and C. 

2. Given a and C. 5. Given h and A. 

3. Given c and A. 6. Given h and C. 
Find the area — 

7. Given a and X 9. Given a and c. 

8. Given a and (7. 10. Given h and (7. 

The General Triangle. 

130. In any triangle ABC, let c denote the base, a and 6 the two 
sides opposite the angles A and B respectively, and h the altitude. 

Then, M= \ ch ; but A — a sin 5. 

Hence, M = \ ac sin B. 



[44] 
Similarly, Jf=^a6sinC, and M=^bcsinA.) 

Hence, the area of a triangle is equal to one-half the product of any 
two of its sides into the sine of the included angle. 

131. A formula may also be found for the area when a side and 
two angles are given, the third angle being then known. 

j? rm, ttt fr sin A b sin C 

From Th. III., a — — — — , c = — — — • 

sin B sin B 



THE RADIUS OF AN INSCRIBED CIRCLE. 93 

Substituting these values of a and c in M = \ac sin B, 

T „ , ,, b 2 sin A sin C 

We have M = — : — 

2 sin B 

Hence, the area of a triangle is equal to the product of the sines of 
any two angles into the square of their included sides, divided by twice 
the sine of the third angle. 

132. A formula may also be derived for the area of a triangle 
when the three sides are given. 

By For. [18], sin B = 2 sin \B cos %B. 

Substituting the values of sin \ B and cos \ B, as given in Art. 76, 

2 



We have sin B = — \/ s (s — a)(s — 6)(s — s). 



ac 



Substituting this value of sin B in [45], 



We have M=\/s(s — a)(s — b)(s — c). [45] 

Hence, to find the area of a triangle when the three sides are given, 
we subtract each side from the half sum of the sides, take the product 
of these differences and the half sum, and extract the square root of 
the product. 

EXERCISES XXVI. 

Find the area of a triangle — 

1. Given a = 20, b = 30, C= 60°. 

2. Given b = 30, c = 40, A = 45°. 

3. Given a =30, c = 40, 5=115°. 

4. Given a = 40, 6 = 80, C = 48°. 

5. Given a = 60, b = 80, c = 150°. 

6. Given b = 100, A = 30°, B = 40°. 

The Radius of an Inscribed Circle. 

133. Let ABC be any triangle whose sides are a, b, and c, and 
r the radius of the inscribed circle •, then dividing the triangle into 
three triangles by drawing lines from the centre to the vertices Of 
the three angles, 



94 



' PLANE TRIGONOMETRY. 



We have M=\ar -\-\br -\-\cr = \r{a-\-b -\- c). 
Let Is denote the sum of the three sides, and 



We have 



M = rs 



whence, 



= M 



[46] 



Hence, the radius of the inscribed circle is equal to the area of the 
triangle divided by one-half the sum of the sides. 

Cor. — Substituting in For. [46] the value of M given in [45], and 
reducing, we have 



\ {8 — a){8 — b){8 — c) . 

\ s 

hence r as used on page 84 is equal to the radius of the inscribed 
circle. 

Radius of a Circumscribed Circle. 



134. Let ABC be circumscribed by a cir- 
cle whose centre is ; and let R denote the 
radius. Draw OB J_ to B C ; then, BD = DC. 

By Geometry, B. III., Th. 18, the angle 
BOC=2A. ! hence, angle BOD = A, and 
BD = R sin BOD, or \ a = R sin A. 




Fig. 43. 



Hence, 

From For. [44], 

Whence, 



sin A = 
R = 



a = 2 R sin A. 

2M 

be 

abc 



4M 



Therefore, the radius of the circumscribed circle is equal to the 
product of the three sides of the triangle divided by four times its 
area. 

Cor. — From Art. 134 we have 

abc 



2R = 



sin A sin B sin C 



Hence, the diameter of the circumscribed circle is equal to the ratio 
of any side of a triangle to the sine of the opposite angle. 



GENERALIZATION OF ANGLES. 95 



EXERCISES XXVII. 

Find the radius of an inscribed circle, 

1. Given a = 4, b = 5, c = 6. 4. Given a = 45, B = 45°, M — 24. 

2. Given a=10, 6=20, J =40°. 5. Given a = 30, C = 60°, Jlf = 40. 

3. Given a=30, 6=35, C=30°. 6. Given a = b = c. 

7. Find the radius of a circumscribed circle in each of the above 
cases. 

8. Find the angles of a right triangle if the hypotenuse is equal 
to four times one of the legs. 

9. Find the legs of a right triangle if the hypotenuse is 12, and one 
acute triangle is twice the other. 

10. Derive a formula for the area of a parallelogram, given two ad- 
jacent sides a and b and the included angle A. 

11. Derive a formula for the area of an isosceles trapezoid, given 
the two parallel sides a and b and acute angle A. 

Generalization of Angles. 

135. We have Art. 54, sin A = 360° + A, or sin A = 2 X 180° 
-f- -A If we add any number of times 360°, as n times 360°, the sine 
is still the same ; hence sin A = 2n X 180° + A or sin A = 2mr + A, 

136. Also sin A = sin (180° — A) or sin (w — A). If we add 
any number of times 360°, as n times 360°, the sine is still the same ; 
hence sin A = sin (n 360° + 180°—^) = sin (2nX 180° + 180° — A) 
= sin {(2n -f- 1) 180° — A} = sin {(2 n + 1) tt — A}. Therefore, if 
A t denotes the general value of an angle A whose sine is a, we have 

A x = 2?i7r -f- A, and A z = (2n-\- 1)tt — A. 

137. From this we infer that if two angles have the same sine, 
either their difference is an even multiple of n, or their sum is an odd 
multiple of if. 

138. Similarly, we may show that if A^ denotes the general value 
of an angle A whose cosine is a, we have 

A 1 = 2nl80°±:A, or A 2 = 2mr =fc A. 

1 . From this it is seen that, if two angles have the same cosine, 
either their sum or their difference must be an even multiple of rr. 



96 PLANE TRIGONOMETRY. 

2. Similarly we may prove that, if two angles have the same tan- 
gent, their difference must be some multiple of tv. 

EXERCISES XXVIII. 

What is the general value of an angle 4 

1. When sin 4 = £? 6. When tan A — 1 ? 

2. When sin A = 1 ? 7. When sec A = 2? 

3. When cos A = 1 ? 8. When cot a A =— 3"|/3? 

4. When sin 2 .4 = £ ? 9. When esc 2 4 = f ? 

5. When tan 2 4 = | ? 10. When tan 4 4 = 9? 

11. Find the general values of A in the equation sin 34 = sin 4 
cos 24. 

Solution. — We have sin (4 + 24) — sin 4 cos 24 = ; whence, 
cos 4 sin 2 4 = 5 hence, either cos 4 = 0, or sin 2 4 = 0. From 
the former we get 4 = some odd multiple of ^7r, and from the latter 
we get 24 = any multiple of it. Hence, both are included in the 
equation 4 = \iiit. 

Find the general value of 4 in the following equations : 

12. cos 4 = cos 2 4. 16. sin 4 4 -\- sin 6 4 = 0. 

13. sin 54 = 16 sin 5 4. 17. tan 4 -f cot 4 = 2. 

14. sin 4 + cos 4 = — — • 18. sec 4 = 2 tan 4. 

V 2 

15. sin 94 — sin 4 = sin 44. 19. esc 4 cot 4 = 2j/3. 

20. sin 4' — cos 4 = 4 sin 4 cos 2 4. 

21. tan (i7r + 4) = 14-sin24. 

Inverse Trigonometric Functions. 

139. The expressions sin 4 = n, cos 4 = n, etc., may also be 
expressed thus : 4 = sin -1 n and 4 = cos -1 n. To read these, notice 
that 

4 = sin -1 n is read, 4 equals the angle whose sine is n. 
A = cos -1 n is read, 4 equals the angle whose cosine is n. 
A = tan -1 n is read, 4 equals the angle whose tangent is n. 

140. These are called inverse trigonometric functions. They 
are often found to be convenient in trigonometry. 

Note. — The student will be careful to notice that in the expression 
sin -1 , the ( — 1) is not to be regarded as an exponent. 



INVERSE TRIGOMETRIC FUNCTIONS. 97 

141. Any relation which exists among trigonometrical functions 
may be expressed by means of the inverse notation. 

EXERCISES XXIX. 

I. What is the value of sin -1 ^? 

Solution. — Evidently sin -1 \ equals 30°, since sin 30° = \. 
Find A, given 

2. A = sin- 1 \ j/2. 6. A = cos^— i) 

3. A = cos- 1 i i/3. 7. A = cot" 1 — | i/3. 

4. A = tan- 1 i/3. 8. ^ = sec - 1 i/2. 

5. A = cos -1 ^. 9. A = esc -1 f i/3. 
10. Given A = tan (cos -1 §), to find the value of A. 

Solution. — This means, what is the tangent of the arc whose 
cosine is f ? Let x denote the angle ; then cos x = § , sin x = \/l — f 
— i l/5 I hence, tan x — ^ j/5 •— § = \ i/5. 

II. Given sina=p, and sin b = q, to express sin (a -f- b) in- 
versely. 



Solution. — Since sin a=.p, cos a= \/l — p 2 ; also cos b = 
l/l — q L ; hence sin (a + 6) — i? l/l — # 2 + <Z V 7 ! — P*- 



Therefore, a + b = sin" 1 (p l/l — 2 2 -f- 2 l/l — j? 2 )- 
Or, sin -1 p -f- sin -1 q '= sin -1 (_p i/l — 2 2 + 2 1 7 1 — _p 2 ). 
Express similarly the inverse functions 

12. Of sin (a — b). 14. Of cos (a — 6). 16. Of tan (a — 6). 

13. Of cos (a + 6). 15. Of tan (a + b). 17. Of cot (a -f- b). 
Prove the following: 

18. tan- 1 ! =-2 tan" 1 i 23. sm- 1 |+sin- 1 1 ^=sm- 1 ff. 



19. sin" 1 ! i/5 + tan" 1 4 = 45°. ^ 

5I ^ ' 3 24. sin -1 = cos -1 i/l — : 

1- 

21. 3 tan- 1 = tan- 1 ^ 



20. 2 tan- 1 == tan" 1 ^~ 

l — 6 " 25. sin- 1 0+cos- 1 = | 



0/y3 

* , 26. tan" 1 + cot" 1 = 5 

22. sin (sin- 1 ^+ cos- 1 !) =1. T 2 

7 



98 PLANE TRIGONOMETRY. 

27. sin I tan -1 — I = cos I cot -1 — I 
V ml \ m I 





\ ml \ m 


28. 


. . 2ajb . 2ab 
sin - ' 1 — tan- 1 • 
a -f- o a — b 


29. 


ta -»i2±i tan _, 1 * 

J/2 — 1 i/-2 i 



Solve the following equations : 

30. sin -1 a; + sin -1 ^ = — • 31. tan -1 2x + tan -1 3 re = — • 

32. sin- 1 2 x — sin" 1 x j/3 = sin- 1 a. 

33. sin 2 x cos" 1 cot 2 tan -1 x = 0. 

Extension of Functions of Two Angles. 
142. In Art. 72 it was shown that the formulas for the sum of 
two angles hold when both angles are obtuse. We now show that 
they are true for all angles, positive or negative. 

1. To do this we will show that they are true when one of the 
angles is increased by 90°. Thus, suppose A 1 = 90° -\- A ; then 
by Art. 52, 

sin (.4 1 -4- B) = sin (90 -f A + B) = cos (A + B). 
And, cos Ol 1 + B) = cos (90 -f- A -f B) = — sin (J. + B). 
Hence, sin (A 1 -\- B) = cos A cos i? — sin A sin 5. 

cos (A 1 -f- B) = — sin A cos B — cos A sin B. 
Now, cos A = sin (90° -f A) = sin ^L 1 . Art. 52. 

sin u4 = — cos (90° -\- A) = — cos A 1 . 
Substituting, sin (A 1 -f- B) = sin J. 1 cos 5 -f- cos A 1 sin 5, 
cos (vl 1 + B) = cos ^l 1 cos B — sin ^l 1 sin B. 

It is thus seen that Formulas [9] and [10] are true when one of 
the angles is increased by 90° ; and in a similar way it may be shown 
that they are true for each increase of either or both angles by 90°, 
and therefore true for the sum of any two angles. 

2. In a similar way it may be shown that the formulas for sin 
(^1 — B) and cos (A — B) are universal. Their universality may 
also be shown by deriving them from sin (A -f- B) and cos (A -f- B), 
which we have just shown are universal. 



EXTENSION OF FUNCTIONS OF TWO ANGLES. 99 

Thus, write (A — B) + B = A. Then by Art. 65, 
sin A = sin [(A — £)+£] = sin (A — B) cos B + cos {A — B) sin B. 
cos A = cos [(A — £)+£] =cos (^ — £) cos J3— sin (^L— 5) sin 5. 
Multiply the first equation by cos B, and the second by sin B, 

sin A cos i? = sin (A — B) cos 2 B -f- cos (J. — B) sin 5 cos 5. 

cos Asm B = — sin (A — B) sin 2 B + cos {A — B) sin B cos 5. 
Subtracting, we have 

sin A cos 5 — cos A sin i? = sin (J. — i?)(sin 2 B -f- cos 2 i?). 
But, sin 2 B -j- cos 2 i? = 1 ; hence, transposing , 

sin (A ■ — B) = sin A cos B — cos A sin B. 
Hence the formula for sin (J. — B), Art. 65, is universal. 

3. In a similar manner, if we multiply the first equation by sin B, 
and the second by cos B, and add the results, an£ reduce, we shall 
obtain 

cos (A — B) = cos A cos B -f- sin A sin B. 

It is thus seen that Formulas [9], [10], [11], and [12] are gen- 
eral for all positive values of A and i4; and therefore all the formulas 
derived from these are also general for all positive values of A 
and A. 

4. Lastly, it may also be shown that these formulas are true for 
any negative values of A and B. 

First, suppose C is negative, and less than A ; then A -J- B becomes 
A — B, and A — B becomes A -\- B, and the formulas are true, as 
already shown. If A is negative, it merely changes the order of 
the letters. 

Second, suppose B is negative and greater than A ; then A — B 
is negative. By Art. 44, 

sin (A — B)-= — sin (B — A) = — (sin B cos A — cos B sin A). 
Whence, sin (A — B) = sin A cos B — cos A sinB. 
Also, cos (A — B) — cos (B — A) = cos B cos A -f sin B sin A. 
Whence, cos (A — B) = cos A cos B -\- sin A sin B. 
The same is also true if A is negative and greater than B. 



100 



PLANE TRIGONOMETRY. 



Third, suppose A and B are both negative ; and let A = — A 1 
&xi&B = — B l . 

sin (A + B) = sin (— A 1 — B 1 ) = — sin (^l 1 + B 1 )- 
= — (sin A 1 cos B l + cos A 1 sin B l ). 
= sin (— .4 1 ) cos (— B l ) + cos (— ^l 1 ) sin (— B l ). 
= sin A cos B -f- cos A sin B. 

5. In the same manner Formulas [10], [11], and [12] may be 
shown to be true when both the angles are negative 5 hence all the 
formulas derived from these are also true. It is thus seen that the 
Formulas [9], [10], etc. are true for every value of A and B, positive 
or negative. 

Application to the Circle. 

143. We hav^ regarded sine, cosine, tangent, etc. of an angle 
as a ratio of the sides of a right triangle, formed by the moving 
radius and its projection. This is the modern 
method of treating trigonometry 5 but the old 
method was to consider these functions as 
lines represented in a circle, «and thus pri- 
marily as functions of arcs. 

144. Thus, in the diagram, by the old 
system, 

1. The line BC is the sine of the arc AB. 

2. The line OC is the cosine of the arc AB. 

3. The line AB is the tangent of the arc AB. 

4. The line OJD is the secant of the arc AB, etc. 

145. In the old system the length of the sine, cosine, tangent, 
etc. depended upon the length of the radius of the circle ; in the new 
system it is a fixed numerical value. 

146. If the radius of the circle is taken as unity, the trigonomet- 
rical functions of angles according to the new system correspond 
with the trigonometrical functions of the arcs of a circle in the old 
system. Thus, denoting the angle AOC, or arc AB, by Z (see Fig. 
44). we have, regarding OB = 1 : 




Fig. 44. 



APPLICATION TO THE CIRCLE. 



101 



(l)sinZ: 
(3)tan^: 



BO. 



(2) COS^: 



BC 
OB 

BC AD Art M . Q ~ 

= = AD. m) secZ- 

OC OA W 



qc_ 

OB 
OB 

OC 



OC. 
OD 



OA 



= OD. 




Fig 45. 



147. This graphic representation of the functions in the old sys- 
tem is for many minds simpler than the more abstract conception of 
ratios, and students should be familiar with it. The following 
definitions are given : 

1. The Sine of an arc is the perpen- 
dicular drawn from its extremity to the 
diameter passing through its origin. 

2. The Cosine of an arc is the dis- 
tance between the foot of the sine of 
the arc and the centre of the circle. 

3. The Tangent of an arc is the 
perpendicular to the radius at its 
origin, and limited by the radius pro- 
duced passing through its extremity. 

4. The Secant of an arc is a line drawn from the centre of the 
circle through the extremity of the arc and limited by a tangent at 
its origin. 

5. The Cotangent and Cosecant are respectively the tangent 
and secant of the complement of the arc. 

148. The functions of arcs terminating in the different quadrants 
are represented in Fig. 45. Thus, 

sin AON is NP; cos AON is OP ; 

tan AON is AT; sec AON is OT; etc. 

sin A ON' is N'P' ; cos AON' is OP' ; 

tan AON' is AT'" ; sec AON' is OT'" ; etc. 

sin AON" is N"P" ; cos AON" is OP" ; 

tan AON" is AT; sec AON" is OT; etc. 

Note. — The student may be required to point out the lines of the cir- 
cle which correspond to the formulas of Tables I., II., and III. 

149. The relation of the values of the trigonometrical functions 
in the two systems is shown as follows. Let R denote the radius 
of the circle ; 



102 PLANE TRIGONOMETRY. 

BC 

Then, sin angle BOC = — — - ; and BC= R X sin angle BOC. 

Also, sin arc AB = BC=^R X sin angle BOC. 

sin of the arc sin 



Hence, sin angle BOC 



radius of circle li 



150. Similar results hold for all the other trigonometrical func- 
tions of the two systems. Hence for any formula of the modern sys- 
tem which involves functions of angles we can readily deduce the 
corresponding formulas in the ancient system depending on the arcs, 
and vice versa. 

Notes. — 1. The modern method was introduced by Dr. Peacock, and 
has almost'entirely superseded the ancient method. 

2. The old definitions give some indications of the origin of the terms 
sine, cosine, etc. The word sine seems to have been derived from the 
Latin word sinus, a bosom. The arc is supposed to represent a bow, and 
thus gets its name; and the string, half of which represents the sine of 
half the arc, would come against the breast of the archer. The words 
tangent and secant are naturally derived from their definitions in Geometry. 

EXERCISES XXX. 

1. Construct the functions of an arc in quadrant II. Show their 
signs. 

2. Construct the functions of an arc in quadrant III. Show their 
signs. 

3. Construct the function of an arc in quadrant IV. Show their 
signs. 

4. Required the signs of the functions of 250° j 320° ; 400° ; 450° : 
600° ; 800°. 

5. Construct the angles less than 360° which have their sine equal 
to ^ ; their cosine equal to ^f ? 

6. Construct the angles less than 270° which have sin A = f ; cos 
A = — f ; tan A = — f ; cot A = $. 

7. Limit the angle when sine and cosine are both positive or both 
negative. When cosine and tangent are both negative or both 
positive. 



APPLICATION TO THE CIRCLE. 



103 



Miscellaneous Exercises. 
Additional Formulas. 

Note. — In 22-29, a, b, c denote the sides of a triangle opposite to the 
respective angles A, B, and C; and S denotes the half sum of the sides. 

5. tan A -}- cot A = 2 esc 2 A . 

6. tan .4 — cot *1 = 2 cot 2^1. 

7. tan 2 A — sin 2 J.=tan 2 ^l sin 2 A 

8. sec 2 .4+csc 2 .4— sec 2 .4 csc 2 .4. 

9. sin 3 A -f- sin A = 2 sin 2 A cos A. 

10. sin 3 A — sin A — 2 cos 2 A sin A. 

11. cos 2 t! 4- cos 4 .4 = 2 cos 3 .4 cos A 
3 tan A — tan 3 .4 



. 


sin 40° + sin 20° = cos 10°. 


2. 


sin 80° — sin 40° = sin 20°. 


3. 


, . . . cos 2 A 
14- sin .4 — . . 
1 — sin A 


4. 


A , L A 1 





12. tan 3 J. 



1 — 3 tan 2 A 



13. tan^ + tan^= sin( / +B ) 
cos A cos B 



14. tan .4 — tan 5 = ^-^ 

c 

15. tan 2 ^4 — tan 2 £ = 



B) 



cos A cos B 
sin 2 A — sin 2 B 



cos 2 A cos 2 B 



16. 
17. 



1 — cos 2 J[ 



tan 2 A. 



1 + cos 2 A 

tan .4 4- tan B sin (A -f- B) 

tan J. — tan B sin (.4 — 5) 



, tan v4 + tan B A D 

18. ! = tan vl tan 5. 

cot A 4- cot 5 



19. 



cos .4 — sin .4 
cos A 4- sin J. 



= sec 2 .4 — tan 2 .4. 



90 cos ^ "4- cos i? cos \ (A — B) 

' sin (^4- B) '" sin$(A + B) 



21. 



cos B — cos A _ sin \ (A — B) 
sm(A + B) ~ o j0 s%(A + B) 



104 PLANE TRIGONOMETRY. 

22. i 1 7 tan ^ = B ec2^ — tan 2 A. 
1 -f- tan A 

sm(A-B) _ (g + b)(a-b) 
' sin (A + B) c 2 

94 sin ^ (vl — #) _ a — b ^ 9 _ cos ^ .4 cos | i? _ 5 

sin | (J. + 7?) c sin | C c 

9 - cos ^ (A — B) a 4- fr 2 o sin ^ J cos ^,- i? s — b 

cos -^ (^4 4- -#) ~ c cos \ C c 

9( . sin ^ A sin ^ B s — c oq cos \ A sin \ B s — a 

sin \ C c cos | C c 

Functions of Special Angles. 
Prove the following, remembering sin 15° = sin (45° — 30°). 

1. sin 15° = V 3 ~ 1 . 3. tan 15° = 2 — i/3. 

2|/2 

2. cosl5° = -^ii 4. cot 15° = 24- t/3. 

2 1/2 

5. Find sin 75° ; cos 75° ; tan 75° j cot 75°. 

6. Find sine of 18°. 

Solution.— Let A = 18°; then 2 A = 36°, and 3 A = 54°, and 
since 36° and 54° are complementary, we have sin 2A = cos 3 A. 
Now, sin 2^4 = 2 sin .4 cos A ; and we can find cos 3.4 = 4 cos 3 
v4 — 3 cos A. Substituting and reducing, we find sin A = £ ( -j/5 — 1) 

= sin 18°. 

Prove the following : 

7. cos 18° = 1/P0 + 2T/B) . g . eos 360 = l+j£5. 

4 4 

9. sin 36°= ^(10-3^3). 
4 

10. sin 9°= yft+vs)-v(5-v5) . 

4 

11. cos 9°= T/CB+^+t/Co-t/S) 
4 



ADDITIONAL EXERCISES. 105 

12 . tan9 o = 6+l£5. 
1/5 — 1 

Note. — Similarly, since 18° — 15° = 3°, we can find the functions of 3°, 
and from this of 6°, etc. 

Find the value of x in the following equations : 

13. sin 2 x = cos x. . 16. tan x + tan (45° -f x) =2. 

14. ■ sin x + cos x = \/1. 17. 2 sin 2 x + sin 2 2 x = 2. 
i 

18. tan (45° — x) + cot (45° — a) = 4. 

19. tan (45° + x) = 3 tan (45° — a). 
25. 6 cot 2 x — 4 cos 2 x = 1. 

21. sin a: + cos a; = ^ j/2. 26. sin 3 £ + sin 2 a; + sin x = 0. 

22. sin x + sin 4 a; = 0. 27. cos 3 x + cos 2a; + cos x =0. 

23. cos 3 x — sin 3 x = \ |/2. 28. sin 2 2 a: — sin 2 x = sin 2 30°. 

24. j/3 sin a: — cos a: = j/2. 

Additional Exercises. 

1. Given sin a; (sin # — cos x) = /- ; find x. 

2. Given tan x + cot a> = 2 ; find a;. 

3. Given sin a; + cos 2 a: = ^ "|/5, to find sin x. 
• 

4. Given 4 sin a? sin 3 x = 1, to find sin x. 

5. Given a sin x -\-b cos x = c, to find sin x. 

6. Given a cos x = tan ar, to find cos x. 

Change to forms for logarithmic computation : 

7. tan x + cot x. 11.1 + tan x tan y. 

8. cot. a; — tan x. 12. 1 — tan x tan y. 

9. tan x + cot y. 13. cot a: cot y + 1. 
10. cot a: — tan y. 14. cot x cot ?/ — 1. 
Demonstrate the following : 



1 — And K * ' 



106 PLANE TRIGONOMETRY. 

16. sin 4 = 4 sin cos — 8 sin 3 cos 0. 

17. cos 4 = 1 — 8 cos' 2 + 8 cos 4 0. 

18. sin 3 = 4 sin sin (60° — 0) sin (60° + 0). 

19. cos 3 = 4 cos cos (60° — 0) cos (60° + 0). 

20. sin (a + 6) sin 3 (a — b) = sin 2 (2 a — b) — sin 2 (2 6 — a). 

21. Find the value of cos (sin -1 ^ -f- cos -1 ^). 

22. Find the value of tan (tan -1 a;+ cot -1 x). 



23. Prove that tan -1 \ -f tan" 1 £ + tan" 1 £ + tan" 1 £ 



7T 



24. Prove that tan (2 tan -1 a) = 2 tan (tan -1 a -f- tan -1 a 3 ). 

25. Prove tan -1 (£ tan 2 a;) -f- tan -1 (cot a;) -j- tan -1 (cot 3 #) = 0. 

1 7T 

26. Find a;, given tan" 1 \ + 2 tan -1 £ + tan" 1 £ + tan- 1 - = -• 

x *± 

27. Prove tan -1 (x — 1) + tan- 1 x + tan" 1 (x + 1) == tan" 1 3 x. 

28. If sec a — esc a = i, prove that a = |- sin -1 f . 

29. If sin A = sin B and cos A = cos B, then either A and B are 
equal, or they differ by some multiple of fou- right angles. 

30. If cos A — cos B and sin A = — sin B, then A -f- B is zero, 
or a multiple of four right angles, positive or negative. 

31. The sum of the tangents of the three angles of a plane triangle 
is equal to their product. 

32. In any plane triangle, cot \ A -f- cot \ B -f- cot £ C = cot ^ 4 
cot | 5 cot \ C. • 

33. In a plane triangle, if b = a sin C and c == a cos -B, then the 
triangle is right-angled at A. 

34. If the angles A, B, and C of a plane triangle are to each other 

as 2, 3, and 4, prove that 2 cos £ vl = ~J~ • 

35. In any plane triangle ABC, if the angle made by a line drawn 
from the vertex C to the middle of the base c is denoted by Z, then 

2 cot Z = cot A — cot B. 



INTRODUCTORY DEFINITIONS, 107 



SPHERICAL TBIGOJSTOMETKY. 



SECTION XII. 

INTRODUCTORY DEFINITIONS. 

151. Spherical Trigonometry treats of the solution 
of spherical triangles. 

152. A Spherical Triangle is a portion of the surface 
of a sphere bounded by three arcs of great circles of the 
sphere. 

153. The sides of a spherical triangle, being arcs of 
great circles, measure the plane angles formed by radii 
of the sphere drawn to the vertices of the triangle. 

154. Each angle of a spherical triangle has the same 
measure as the dihedral angle included by the planes of 
its sides. 

155. The sides of a spherical triangle may have any 
values from 0° to 360° ; but in this treatise only sides less 
than 180° will be considered. The angles may have any 
values from 0° to 180°. 

156. If two parts of a spherical triangle are both 
greater or both less than 90°, they # are said to be in the 
same quadrant; but if one part is greater and the other 
less than 90°, they are said to be in different quadrants. 

157. Spherical triangles are divided into two general 
classes'— right spherical triangles and oblique spherical triangles, 



108 SPHERICAL TRIGONOMETRY. 

the same as plane triangles. A right spherical triangle 
may have one, two; or even three right angles. 

158. A spherical triangle having two right angles is 
called a bi-rectangular triangle. A spherical triangle having 
three right angles is called a tri-rectangular triangle. A 
spherical triangle having one or more sides equal to a 
quadrant is called a quadrantal triangle. 

159. The nature of a spherical triangle will be seen by 
examining the diagram in the margin. The side AB 
measures the plane angle 
AOB; the side BQ measures 
the plane angle BOC, and 
the side AC measures the 
plane angle AOC. 

The spherical angle B is 
measured by the dihedral Fig. 46. 

angle formed by the two planes AOB and COB ; the 
spherical angle C is measured by the dihedral angle 
formed by the two planes AOC and BOC, etc. 

160. It will be remembered, as shown in Geometry, 
that in every spherical triangle we have the following 
truths : 

1. The sum of the sides is less than 360°. 

2. The sum of the angles is greater than 180° and less than 
540°. 

3. If two angles of a spherical triangle are equal, the opposite 
sides are equal. 

4. If one angle of a spherical triangle is greater than an- 
other, the side opposite the greater angle is greater than the side 
opposite the less angle— and conversely. 




INTRODUCTORY DEFINITIONS. 



109 



5. The sides and angles of any spherical triangle are re- 
spectively the supplements of the angles and sides of the polar 
triangle. 

161. Thus, if the angles of a 
spherical triangle are denoted by 
A, B, C, and the sides opposite these 
angles respectively by a, b, c, and the 
corresponding angles and sides of the 
polar triangle by A\ B', C, a', b\ and 




<?', then we shall have the following 



Fig. 47. 



tions : 

A = 180° - 


-a'. 


C= 180° -c'. 


B'=lS0° — b. 


B = 180° - 


-V. 


A' =180° — a. 


C'=180° — c. 



EXERCISES XXXI. 

1. If the angles of a spherical triangle are 50°, 60°, and 85°, what 
are the sides of its polar triangle ? 

2. The sides of a spherical triangle are 75°, 97°, and 115°; what 
are the angles of the polar triangles ? 

3. What kind of a triangle is the polar triangle of a quadrantal 
triangle? Ans. A right triangle. 

4. If a triangle has three right angles, what is the length qf the 
sides of the triangle ? Ans. Quadrants. 

5. Prove that if a triangle has two right angles, the sides opposite 
these angles are quadrants, and the side opposite the third angle 
measures that angle. 

6. Find the length of the sides of the polar triangle in Ex. 1, in 
units of length, if the diameter of the sphere is 8 units. 



110 SPHERICAL TRIGONOMETRY. 

SECTION XIII. 

THE RIGHT SPHERICAL TRIANGLE. 
Fundamental Formulas. 

162. We proceed first to find the relation of the 
functions of the sides and angles of a right spherical 
triangle. 

Let ABC be a right spherical .? 

triangle, right angled at C; JZf \ \? 

and let be the centre of the s' ; V y \ 

sphere. Denote the angles by 6> <T /-^ 4c 

the letters A, B, and C, and ^""l^-^_ \ A 

their opposite sides by a, b, a 

, Fig. 48. 

and c. 

Draw OA, OB, and OC, each of which will be the radius 
of the sphere. From F draw FE _L to OA, and from E 
erect ED J_ to OA, and draw FD ; then the angle FED 
will measure the dihedral angle whose edge is OA ; and 
angle FED = angle A. 

The plane FDE is _L to the plane AOC (B. VI. Th. 21) ; 
hence FD, the intersection of the planes FDE and FOC, 
is JL to the plane AOC (B. VI. Th. 24); therefore, FD 
is _L to OC and DE. 

Now from the principles of Plane Trigonometry, and 
changing the form of some of the expressions by multi- 
plying and dividing by the same quantity, we have the 
following results : 

OE OE OD ., , FA „ 

of = ~6d x of ; 1S ' cos c == cos a cos E '1 



FUNDAMENTAL FORMULAS. 


Ill 


FD FD FE m 
0F~ FE X OF' 


that is, sin a == sin A sin c. 


1 
k48] 


Similarly, 


sin b = sin B sin c. 


J 


Again, 






BE DE OE 
EF~ 0E X EF ] 


or cos A = tan b cot c. 


[[49] 


Similarly, 


cos B = tan a cot c. 


J 


Again, 







ED ED I'D . . , . , 

tttt = -77^ X 7777 ; or sin b = cot A tan a. 

OD FD 0D> l [50] 

Similarly, sin a = cot B tan b. 

Taking the product of the two formulas [50], we have 

sin a sin b , . . _, 

7 — 7 = cot A cot B. 

tan a tan 0. 

Whence [47], cos c = cot A cot B. [51] 

Multiply the first formula in [48] by the second in [49], 
we have 

sin a cos B = sin A sin c tan a cot c. 

, Tr , _, . . tan a . , . . cos c 

\\ hence, cos B = sin A X sin c cot c = sm A 

sin a cos a 

Or, [47], cos B = sin A cos b. 



[52] 
Similarly, cos A = sin B cos a. 

163. In deriving the formulas under Art. 162, it was 
assumed in the construction of the figure that all the parts 
of the triangle, except the right angle, are less than 90°. 
The formulas are, however, true for any right spherical 
triangle, as is readily seen. 



112 



SPHERICAL TRIGONOMETRY. 



OE OD 
0F~ OD X OF 



Suppose one leg a to be greater 
than 90°. Then construct a fig- 
ure (Fig. 49), as in Art. 162. 

.Now, 

Or, 

cos (180°— c)=cos b cos (180°— a) 

Whence, cos c=cos a cos b ; 
and this is the same as Formula [47J. 

Again, suppose that both legs a and b are greater than 90°. 
Construct the figure as before (see Fig. 50). 
OE 




Fig. 49. 



Then, 



Or, 



_ OE OD 
OF" OD X OF' 



cos c = cos (180° - b) cos (180° —a). 
Whence, cos c = cos a cos b. 

and this is the same as Formula [47]. 
Therefore the formulas of Art. 162 
are universally true. 




Fig 50. 



EXERCISES XXXII. 

1. If c = 90°, what may be inferred in respect to the other parts? 
Solution. — In For. [47] cos c = cos a cos b. If c = 90°, cos c = ; 

hence, either cos a or cos b is 0, and either a or b is equal to 90°. If 
a = 90°, then A = 90°, and B=b. If b = 90°, then B = 90°, and 
A = a. • 

2. If a = 90°, what may be inferred in respect to the other parts ? 
If a = 90° and c = 90°? If a = 90° and 6 = 90°? 

3. What may be inferred in respect to the other parts if b = 90° ? 
If.a = ^? If b = B, orc= C? 

4. What will each of the formulas in Art. 162 become when ap- 
plied to the polar triangle ? 



NAPIER'S RULES. 



113 



Formulas of Plane and Spherical Compared. 

164. The six formulas of Art. 162, comprising ten 
equations, enable us to solve every case of right spherical 
triangles. Put in another form, as below, they may be 
remembered by their analogy to the corresponding formu- 
las for plane triangles : 



In Plane Right Triangle. 



sin A = 



cos A = 



tan^. = 



sin B = - 
c 



cos B — 



tani? = 



sin A = cos B. sin B 
c* = a* + b\ 
1 = cot A cot B. 



a 

cos A. 



In Spherical Right Triangle. 
sin J. 



sin a . _, sin b 

sm5 = — 

sin c sin c 



. tan b „ tan a 

cos^4=- • cosi5 = : 

tan c tan c 



tan A 



tana . „ tanfr 
tan 5 = 



sin b 



sin a 

cos A 



. . cosi? . 

sin J. = 7- sin i* = 

cos 6> cos a 

cos c = cos a cos &. 

cos c= cot A cot 5. 



Napier's Rules. 

165. The ten formulas of Art. 162, by a very ingenious 
device, may all be embraced in two general rules, easily 
remembered and applied. These rules are due to Baron 
Napier, the distinguished inventor of logarithms. 

166. In this device, five of the parts of the triangle are 
considered, the two sides about the right angle, the comple- 
ments of their opposite angles, and the complement of the 
hypotenuse. These are called Napier's Circular Parts. 

These parts are represented thus : a, b, co. A, co. B, 
and co. C. Notice that co. A equals 90° — A, etc. 

167. Any one of these parts may be taken as the middle 

8 




114 SPHERICAL TRIGONOMETRY. 

part; and then the two parts adjacent to it are called 
adjacent parts, and those which are separated fro.m it are 
called opposite parts. 

co. B 

Thus, if CO. O is taken as the middle 

part, then co. A and co. B are adjacent 
^ parts, and a and b are opposite parts, as 
is seen in the figure. eo ^ 

It will be noticed that the right angle 
does not enter as one of the parts, and 
that the two sides including it are regarded as adjacent. 

168. The two rules of Napier are as follows : 

Rule I. The sine of the middle part is equal to the product 
of the tangents of the adjacent parts. 

Rule II. The sine of the middle part is equal to the product 
of the cosines of the opposite parts. 

Note. — It will aid the memory to notice that the vowel o occurs 
in cosine and opposite, while a occurs in tangent and adjacent. 

169. The correctness of these rules may be shown by^ 
taking each of the five parts as the middle part, and com- 
paring the resulting equations with the formulas of Art. 
162. 

Thus, let co. c (see Fig. 51) be taken as the middle part ; 
then co. A and co. B are the adjacent parts, and a and b 
are the opposite parts. Then by Napier's Rules we have 

sin (co. c) = tan (co. A) tan (co. B). 
Whence, cos c = cot A cot B. 
Also, sin (co. c) = cos a cos b. 

Whence, cos c = cos a cos &. 
These results, it will be seen, correspond with Formulas 



THE AMBIGUOUS CASES. 115 

[51] and [47] ; and in a similar manner all the formulas 
may be derived from the two rules. 

Note. — The rules were originally derived from the formulas, and 
may be so derived by substituting for A, B, and C in the formulas, 
their complements. 

EXERCISES XXXIII. 

1. Derive Formulas [48] from Napier's Rules. 

2. Derive Formulas [50] from Napier's Rules. 

3. Derive Formulas [49] and [52] from Napier's Rules. 

4. If we take for the five parts of the triangle the hypotenuse, 
the two oblique angles, and the complements of the legs, what for- 
mulas will Napier's Rules give? ■ 

5. On this supposition, what rules should we have to give the 
same results as Napier's Rules? 

Note. — The rules thus derived are known as Manduifs Rules. 

The Ambiguous Cases. 

170. In applying Napier's Rules, or the formulas of 
Art. 162, where the part sought is to be determined by 
the sine — the same sine corresponds to two different angles 
or arcs, supplements of each other — it becomes necessary 
to discover such a relation between the parts as will enable 
us to determine which of the two angles or arcs is to be 
taken. 

171. For this purpose we shall prove the following 
principles : 

Prin. 1. In a right spherical triangle, a side and its opposite 

angle are always in the same quadrant. 

For we have [52], 

. cos B 

sin A — • 

cos b 



116 SPHERICAL TRIGONOMETRY. 

Now A is always less than 180° ; hence sin A is always 
plus ; therefore cos B and cos b must always have the same 
sign, and hence must both be greater or both less than 90°. 

Prin. 2. If the tivo sides of a right spherical triangle, in- 
cluding the right angle, are in the same quadrant, the hypote- 
nuse is less than 90° ; but if the two sides are in different quad- 
rants, the hypotenuse is greater than 90°. 

For we have [47], 

cos c = cos a cos b. 

Now, if cos a and cos b have the same sign, cos c is 
positive, and hence c is less than 90° ; but if cos a and 
cos b have different signs, cos c is negative, and hence c 
is greater than 90.° 

Note. — These two principles enable us to determine the nature of 
the part to be found in every case, except when an oblique angle 
and an opposite side are given to find the other parts. In that case 
there may be two solutions, one solution, or no solution, as will be 
shown in the treatment of the case. 

Solution of Right Spherical Triangles. 

172. In the solution of right spherical triangles there 
are six cases, as follows. Given, 

1. The two legs. 3. The hypotenuse and one leg. 

2. The two angles. 4. The hypotenuse and one angle. 

5. One leg and its adjacent angle. 

6. One leg and its opposite angle. 

173. In solving these several cases the formulas given 
under Art. 162 may be taken from the book, or these 
formulas may be readily derived from Napier's Rules. 

1. In applying Napier's Kules to obtain the formulas, it will be 
readily seen which of the three parts — the two given and the one 



SOLUTION OF RIGHT SPHERICAL TRIANGLES. 117 



co. B 



co. A 




required — is to be taken as the middle part. Thus, if the three 
parts are all adjacent to one another, the middle one of the three 
is the middle part, and the other two are adjacent parts ; if one is 
separated from the other two parts, then the part which stands by 
itself is the middle part, and the other two parts are opposite parts. 

2. Thus, suppose we have a and b given to find the other parts, 
then to find c we write the terms, a, 6, 
co. c, and since co. c is separated from a 
and b (see Fig. 52), we take co. c for the 
middle part, and have by Rule I., 
sin (co. c) = cos a cos b, 
or, cos c = cos a cos b. 

To find A we write a, b, and co. A, 
and since the terms are not separated (see Fig. 52), we take b for 
the middle part, and by Rule II. have 

sin b = tan a tan (co. A) = tan a cot A. 

Whence, cot A= cot a sin /;. 

Students are advised to derive the formulas from Napier's Rules. 

Case I. 

174. Given the two legs a and b of the triangle 
ABC 

1. In the spherical triangle ABC, right angled at C, 
a = 59° 38' and b = 48° 24' ; find A, B, and c. 



Fig. 52. 



Solution. — The formulas for 
the solution, derived by Napier's 
Rules or taken from Art. 162, are, 
cos c = cos a cos b. [47] 
cot A = cot a sin b. [50] 
cot B = sin a cot b. [50] 
Here C, A, and B are deter- 
mined by the cosine, and there 
is no ambiguity. 



Operation. 
log cos a (59° 380 = 9.703749 
log cos b (48° 240 = 9.822120 
log cos c = 9.525869 

c = 70° 23 / 20" 

log cot a (59° 380 = 3.767834 

log sin b (48° 240 = 9-873784 

log cot A = 9.641618 

A = 66° 20 7 23" 

Similarly, B = 52° 32' 48" 



118 



SPHERICAL TRIGONOMETRY. 



EXERCISES XXXIV. 

2. In the right spherical triangle, given a = 75° 15' and 
b = 120° 15' ; find A = 77° 11' 14", B = 119° 25' 17", c = 
97° 22' 9". 

3. In the right spherical triangle ABC, given a = 155° 
27' 54", and b = 29° 46' 8" ; find c = 142° 9' 13", A = 137° 
24' 21", B = 54° 1' 16". 

Case II. 

• 175. Given the two oblique angles A and B of the 
triangle ABC. 

1. In the spherical triangle, right angled at 0, A = 62° 
15', and B = 56° 30'; find «, b, and c. 

Operation. 
log cot A (62° 150 = 9.721089 
log cot B (56° 3CK) = 9.820783 
log cos c = 9.541872 

^z=69°37 / 14" 

log cos A (02° If/) = 9-668027 

log sin B (56° 300 - 9.921107 

log cos a ?= 9.746920 

a = 56° 3' 25" 

Similarly, 6 = 51° 24 / 56" 



Solution. — The formulas for 

the solution, taken from Art. 1G2 

or derived by Napier s Rules are, 

cos c = cot A cot B 

cos A = cos a sin B 

cos B = cos 6 sin yl 

From the second and third we 

have, 

cos a = cos ,4 -!- sin 7? 

cos 6 = cos B -r sin J, 

which we use to find a and 6. 



EXERCISES XXXV. 

2. In the right spherical triangle ABC, given ,4 = 69° 
20', B = 58° 16'; find a = 65° 28' 58", b = 55° 47' 46", 



SOLUTION OF BIGHT SPHERICAL TRIANGLES. 119 

3. In the right spherical triangle A B C, given A = 47° 
13' 43", B = 126° 40' 24" ; find a =32° 08' 56", b = 144° 
27' 03", c = 133° 32' 26". 

Case III. 
176. Given the hypotenuse c and either leg a 
or b. 

1. In the spherical triangle ABC, right angled at C, c — 
56° 13', a = 48° 30' ; find A, B, and &. 

Operation. 



Solution. — The formulas for 
the solution taken from Art. 162, 
or derived by Napier's Rules, are, 
cos c = cos a cos b 
sin a = sin A sin c 
cos 5 = tan a cot c 
Whence, cos b = cos c -f- cos a 
and sin ^4 = sin a —• sin c 



log cos c (56° 130 = 9-745117 
log cos a (48° 300 = 9-821265 
log cos b = 9.923852 

b = 32° 56' 49" 

log sin a (48° 300 = 9.874456 

log sin c (56° 130 = 9.919677 

log sin A = 9.954779 

^ = 64°18 / 17" 

Similarly, B = 40° 52 r 14" 



Note. — Two angles correspond to sin A, but since a is less than 
90°, the angle A must also be less than 90° (Art. 170). 



EXERCISES XXXVI. 

2. In the right spherical triangle ABC, given b — 37° 
48' and c = 66° 32' ; find B = 41° 55' 34", A = 70° 19' 18", 
and a = 59° 44' 13". 

3. In the right spherical triangle ABC, given a = 95° 
22' 30", c = 91° 42" ; find A =^ 95° 6', B = 71° 36' 45", and 
b = 71° 32' 12". 



120 



SPHERICAL TRIGONOMETRY. 



Case IV. 

177. Given the hypotenuse c and either angle A 
or B. 

1. Given c = 86° 50' and A 



Solution. — The formulas for 
the solution taken from Art. 162, 
or derived by Napier's Rules, are, 
sin a = sin A sin c 
tan b = cos A tan c 
cos c = cot A cot B 
Whence, cot B = cos c tan A 



58° 3<J; find a, &, and £. 

log sin A (58° 300 = 9-930766 
log sin c (86° 500 = 9.999336 
log sin a = 9.930102 

a = 58° 21 / 27" 
log cos A (58° 300 = 9.718085 
log tan c (86° 500 = 1 1-257078 
log tan 6 =10.975163 

b = 83° 57" 21" 
Similarly, 5 = 84° 50' 56" 



EXERCISES XXXVII. 

2. Given c = 115° 35' 20", and 5 = 110° 26' 30"; find 
a == 36° 6' 13", & = 122° 18' 54", and A = 40° 47' 35". 

3. Given c = 70° 23' 42" and A = 66° 20' 40" ; find a = 
59° 38' 26", & = 48° 24' 15", and B = 52° 32' 55". 

Note. — In Ex. 1, two values of a correspond to sin a; but by 
Prin. I., a must be less than 90°, since A is less than 90°. Similarly, 
in Ex. 2, b must be greater than 90°. 

Case V. 
178. Given one leg a and its adjacent angle B. 
1. Given a = 102° 30' and B= 43° 24'; to find b, c, 
and A. 



Solution. — The formulas for 

the solution derived by Napier's 

Rules or taken from Art. 1 62, are, 

tan b = sin a tan B 

cot c = cot a cos B 

cos A = cos a sin B 



log sin a (102° 300 = 9.989582 
log tan B (43° 24Q = 9.975732 
log tan b = 9.965314 

^ __ 42° 42' 52 // 
Similarly, c = 99° 9 / 2" 
and A = 98° 33' 9" 



SOLUTION OF BIGHT SPHERICAL TRIANGLES. 121 
EXERCISES XXXVIII. 

2. Given b = 42° 40 7 24", and A == 116° 36' 20"; find 
a = 126° 27' 47", c = 115° 54' 35", and B = 48° 54'. 

3. Given a = 29° 46' 8", and B = 137° 24' 21"; find A = 
54° 1' 16", b = 155° 27' 54", and c = 142° 9' 13". 

Note. — In Ex. 1, since cot a is negative, cot c is negative, and 
hence c is greater than 90°. In Ex. 2, a is greater than 90°, since A 
is greater than 90°. 

Case VI. 

179. Given one leg a and its opposite angle A. 

1. Given a= 110° 32' 25" and A = 98° 48' 50"; find 6, 
c, and B. 

Solution. — The formulas for 
the solution are 

sin b = tan a cot A 
sin c = sin a — sin A 
sin 5 = cos A -r cos a 



log tan a (110° 82' 25") == 10.426332 
log cot A (98° 48' 50") = 9.190490 
log sin 6 = 9.616822 

b = 24° 26' 44" or 155° 33 / 16" 
Similarly, 

c = 71° 22 / 23" or 108° 37' 37" 
And, B = 25° 53 / 38" or 154° 6 / 22" 



Note. — In this case, since all the required parts are 
determined by their sines, there are always two solu- 
tions. Thus, if in the triangle ABC, AB and AC 
are produced to meet in -4', ABA / and ACA / are 
semi-circumferences, and the angle A — A / . The 
two triangles, ABC and A'BC, both have the two 
given parts a and A : but ?/, c', and 2?' in the second 
triangle are respectively the supplements of 6, c, and 
B in the first triangle. 




Fig. 53. 



122 



SPHERICAL TRIGONOMETRY. 



EXERCISES XXXIX. 

2. Given, A = 102° and a = 120° ; find the other parts! 
Ans. b = 21° 36' 08" "] f b = 158° 23' 52" 

c = 62° 17' 51" I or J c = 117° 42' 09" 
B = 24° 34' 16" J [ £ = 155° 25' 44" 

3. Given B = 80° and b = 75° ; solve the triangle. 
Ans. a = 41° 09' 18" ^ f a= 138° 50' 52' 



c= 78° 45' 45" 
^=42° 08' 18" 



or 



c = 101° 14' 15" 
4=137° 51' 42' 



Quadrantal Spherical Triangles. 

1 80. A Quadrantal Spherical Triangle is one in which 
one side is equal to 90°. It is the polar triangle of some 
right spherical triangle. 

181. To solve a quadrantal spherical triangle we pass 
to its polar triangle by subtracting each side and angle 
from 180°. The resulting polar triangle will be right 
angled, and may be solved as already explained. The 
parts of the given triangle may then be found by sub- 
tracting the parts of the polar triangle from 180.° 

EXERCISES XL. 

1. Given the quadrantal triangle ABC, in which c — 90°, 
£=42° 10' and C= 115° 20'. 

Solution. — Passing to the polar triangle 
A'B'C, we have C = 90°, c' = 64° 40', 
and 1/ = 137° 50'. 

Solving this triangle by the method for 
right triangles, we find A' = 115° 23' 20", 
B' = 132° 2 / 13", and a* = 125° 15 / 3G". 
Subtracting each of these from 180°, we find the required parts of 




Fig. 54. 



FUNDAMENTAL FORMULAS. 123 

the quadrantal triangle are BC '= 64° 36' 40", AC = 47° 57' 4*7", 
and J. = 54° 44' 24". 

2. Let ^4i>(7 be a quadrantal triangle in which c = 90°, 
4 = 75° 42', and 6= 18° 37"; find (7= 103° 34' 49", J5 = 
18° 04' 40", a = 85° 28' 39". 

3. In the spherical triangle ABC, given a, b, and c, each 
equal to 90°, to find the angles. 

182. An Isosceles Triangle is readily solved by divid- 
ing it into two right triangles by drawing an arc of a great 
circle from the vertex perpendicular to the base. 



SECTION XIV. 

THE OBLIQUE SPHERICAL TRIANGLE, 
Fundamental Formulas. 

183. We now proceed to find the relation of the func- 
tions of the sides and angles of an oblique spherical 
triangle. 

I. To find the relation of the sines of the sides and 
angles. 

184. Let ABC, Fig. 54, be an 
oblique spherical triangle, A, B, C 
its three angles and a, b, c its three 
sides. 

From C draw an arc CD of a 

great circle perpendicular to the 

side AB, meeting; AB in D: and 

° Fig. 55. 

denote CD by p. 




124 



SPHERIC A L TRIG ONOMETR Y. 



[53] 



In the right triangles BCD and ^ CD, we have (Art. 162), 

sin p = sin a sin B 

sin p = sin b sin A 
Hence, sin a sin B = sin b sin A 

Similarly, sin a sin C = sin c sin A 
And sin b sin C = sin c sin B 

These equations may be written in the form of propor- 
tions; as, 

sin a : sin b = sin A : sin B. 

Hence, we have the following theorem : 

1. The sines of the sides of a spherical triangle are propor- 
tional to the sines of their opposite angles. 

185. If in Fig. 54, the perpendicular CD cuts the side 
AB produced, we must have in place of sin A, sin B, or 
sin 0, sin (180° - A), sin (180° - B) or sin (180° - C). 
But these sines are equal to sin A, sin B, and sin C, re- 
spectively (Art. 52) ; hence the formulas [53] are true for 
all cases. 

II. To find an expression for 
the cosines of the sides. 

186. In the triangle ABC, CD 
being perpendicular to the base as 
before, let AD = m and BD = n. 

Now, in the right triangle BCD 
we have (Art. 162), 

cos a = cos p cos n = cos p cos (c — m) Fl §- 56 

Or, 
cos a =-• cos p cos c cos m + cos p sin e sin m. 




FUNDAMENTAL FORMULAS. 125 

But, Art. 162, cos p cos m = cos b. 
Whence, cos p = cos b sec m. 

And, cos p sin m = cos & tan m. 

• Or, Art. 162, = cos b tan & cos A. 

= sin & cos A 

Substituting these values of cos p cos m and cos p sin m 
in the second expression above, we obtain 

cos a = cos b cos c -f sin b sin c cos A 1 
Similarly, cos b = cos a cos c -f sin a sin c cos B I [54] 
And, cos c = cos a cos b -f- sin a sin b cos C J 

These formulas give the following theorem : 
2. In any spherical triangle the cosine of each side is equal 
to the product of the cosines of the other two sides plus the product 
of the sines of these sides and the cosine of the included angle. 

III. To find an expression for the cosines of the 
angles. 

187. Let AB'C be the polar triangle of ABC, and 
denote its angles by A', B\ and C, and its sides by d, &', c v . 
Then from Art. 186, we have 

cos d = cos V cos e' -f sin V sin d cos A f . 

Now by Art. 161, 

A' = 180° — a, B'= 180° — &, <7' = 180° — C. 

d==lS0° — A, b'=180°—B, c '=180°— C. 
Substituting these values in the first formula [47], we have 

— cos A = ( — cos B) (— cos (7) — sin B sin C cos a. 
Whence by changing the signs, we have 

cos A=sin B sin C cos a — cos B cos C ^| 
Similarly, cos B=sin C sin A cos b — cos C cos A I [55] 
And, cos C=sin A sin B cos c— cos A cos B J 



126 SPHERICAL TRIGONOMETRY. 

188. In this way, by means of the polar triangle, any 
formula of a spherical triangle may be transformed into 
another in which angles take the place of sides and sides 
of angles. 

189. By making one of the angles of the spherical tri- 
angle a right angle, all the formulas of a right sj)herical 
triangle, given in Art. 162, can be obtained from the form- 
ulas of an oblique spherical triangle. 

EXERCISES XLI. 

1. Show what formulas may be derived from [53] by making A = 
90° ; making B = 90° ; making C = 90°. 

2. Show what formulas may be derived from [53] by making a = 
90° ; making A = B = 90° ; making a ■ = b = 90°. 

3. What formulas may be .derived from [54] by making A = 90° ; 
5 = 90°; C=90° ■ a = 90°; 6 = 90°; c=90°? 

IV. To find expressions for half angles and sides. 

190. From the first equation of Art. 186, we deduce 

A cos a — cos b cos c 

COS A = : ; — : • 

sin b sin c 
Subtracting this equation from unity, we have 
H . sin & sin c + cos b cos c — cos a 

1 — COS A — : — — ~ 

sin b sin c 

Substituting values as given in Arts. 69 and 70, 

. , , . sin i (a + & — e) sin \ ( a — b -f c) 

sin 2 J A = : — 7—. — "— * 

z sin b sm c 

Let s = \ (a 4- b -f c) ; then -J- (a + b — e) = s — c, and 
\ (a — b -f c) = s — b. 
Substituting, we have 



FUNDAMENTAL FORMULAS. 127 

sin (g — b) sin (s — c) 
sin & sin c 

sin (a — g) sin (s — a) 



sin * ^. - 

^ sin 6 sin c 



Similarly, sin 2 4 B = 

J ' ^ * sine sin a 

And s inHC= Sin(8 T a)sill l S - 6 > 

sm a sm o 



[56] 



191. If we add unity in Art. 190, and reduce as before, 
we may derive the following formulas : 



, , A sin s (sm s — a) 
COS 2 I A = r^i — : 

sm o sm g 

o , -r, sin s (sin s — b) 

cos IB — r- 1 : ■ 

* sm c sm a 



„ , ~ sm s sm (s — c) 

COS 2 A C = : -: — ; — 

* sm a sm b 



[57] 



192. Dividing the corresponding formulas of Arts. 190 
and 191, we have, For. [2], 



tanH^ = Sin . (S ~ 6)S ; n(5 ^ £ - ) - 

sm s sm (s — a) 

^ ■rinfr-«)yi(»-a) 

sm s sm (s — b) 

sin (g — a) sin (s — &) 
sin s sin (s — c) 



tan 2 | = 



[58] 



193. Again, from the first equation of [55], we have 
cos B cos Q + cos A 



cos a = 



sin i? sin C 



Whence, 1 - cos a = - cos ^ + c p os ( B + C) ■ 

sm B sm C T 

„ . 21 cos±(A + B+ C) cos ±(B+C— A) 

Hence, sm 2 -Ja=- ^ — ! — ^— : — 77 — - — ■ ~ 

2 sm B sm C 



128 SPHERICAL TRIGONOMETRY. 

Now let s = i (A + B + C) ; then, J (£ + C— 4) = (s-,4) ; 

, Tr , . 21 — cos Scos(S— A) 

Whence, sin 4- a = : — r— : — — — - • 

' 2 sin i? sin (7 

<-,..,, . 2 ii _ cos£cos(£ — 5) 

Similarly, sin 2 4- & == : — ^ . . > [59] 

J ' 2 sin sm .4 

. , . „ , — cos S cos (S — 0) 

And, Sin 2 f C = r : — r — z: • 

A sm A sm B 

194. In a similar manner we may find the following : 



! 2 JL 



cos (S-B) cos (S-C) 



sm i* sm C 



2 . , cos (S — C)cos(S-A) 
C0S * b = sinCs hii " ' WW 



_ cos (£-.4) cos (S— J?) 
sin J. sin i? 



cos 2 4- c = 



195. And from [59] and [60] we derive, by For. [2], 

, 9 , — cos S cos (S — A) 

tan 2 4- a = 



2 w cosOSf-£)cos(£-C) 

_ — cos S cos Qg — B) 
xan 2»- c@s( ^__ C)cog( ^__^ ) 

_ - cos S cos QS-C) 
xan 2 ~ cos (5— -4) cos (£— £) 



[61] 



Notes. — 1. The second members of Formulas [59] and [61] must 
be essentially positive, though their algebraic sign is negative ; for 
since 2 S> 180°, S^> 90° and cos S is negative ; hence, — cos S is 
positive. Also, the positive sign must be given to the radical, since 

2 

— is less than a right angle. 



2. Formulas [59], [60], and [61] might have been deduced by 
applying Formulas [56], [57], and [58] to the polar triangle. 



GAUSS'S EQUATIONS. 129 

Gauss's Equations. 

196. From For. [9] we have 

sin J (A + B) = sin \ A cos ^ B + cos J ^4 sin J i?. 

Substituting the values of sin J A, cos J I>, cos ^ A, and 
sin \ B, derived from [56] and [57], and reducing by com- 
bining factors and extracting root, we have 

±(A+E)= / sin ( g ~ & ) sm ( g — c ) x / sin s sin (s— 6) 
2 \ sin b sin c \ sin c sin a 



sin 



hm«am(.-a) J 
\ sm o sin c \ 



/sin s sin (s — a) ^ f sin (s— c)sin (s— a) 
sin c sin a 



_ sin (.9 — a) + sin (s— 5) / sin s sin (s — c) 
sin c \ sin a sin 5 

Now, sin c = 2 sin ! c cos | c [17], and sin (s — a) -f sin 
(s — 5) = 2 sin |- c cos | (6 — a) [31] ; and the quantity 
under the radical equals cos J C [57], hence, 

■ " i / a- , ™ 2 sin 4- c cos -I Cb — a) , n 

sm \ (A + B) = TT^—i i x cos i a 

2 sm -J c cos \ c 

Cancelling, multiplying by cos -J c, and reducing, 
sin \ {A + B) cos J c = cos ^ (a — Z>) cos | 0. 
Operating in the same, way with the values of 
sin \ (A — B), cos l(A + B), and cos i (A — 5), ■ 
we have the four equations, 

sin ^(A-{- B) cos^c = cos J (a — 6) cos -J- C 
cos -J- (^4 -f 5) cos \ c = cos J (a + 6) sin J C. 
sin £ (^4 — B) sin -J- c = sin J (a — 6) cos \ C. 
cos |(^4 — B) sin f c = sin -J- (a + &) sin J C 



y [62] 



K63] 



130 SPHERICAL TRIGONOMETRY. 

These four formulas are called Gauss's Equations, though, 
as Todhunter remarks, they are realty due to Delambre. 

Napier's Analogies. 

197. By dividing the first of Gauss's Equations by the 
second, the third by the fourth, the fourth by the second, 
and the third by the first, we obtain the following equa- 
tions : 

, 1,4,-™ cos \ ( a — V) , , n 

tan I (A -f E) = \- f — r~d cot i <?• 

2 v J cos \ (a+b) 2 

± i / a ™ sin -i- (« — b) 1 n 
tan \ (A — E) = - — r~7 — TT^ cot i c - 

w , _ . cos 4- M— B) , . 
tanHa + &) = C0 4^ +jB) tan|c. 

tanJ(«-J) = ^fg=f tanlc 

198. Writing these equations in the form of proportions, 
we have, 

sin -J- (a +6) : sin -J (a— 6) = cot J C\ tan -J- {A — B). 

cosi(a + &):cosi(a— &) = cot J (7: tan 1(^+5). 

r [64] 
sin i (J. + B) : sin } (A—B) = tan J c : tan | (a— 6). 

cos \ (^4+£) : cos i (A—B) = tan J c : tan \ (a+&). 

These proportions are called, from their inventor, Napier's 
Analogies. 

Note. — As is seen, there is a very intimate relation betAveen Gauss's 
Equations and Napier's Analogies. We have derived the Analogies 
from the Equations ; but the Analogies may be derived by an inde- 
pendent process, and the Equations deduced from the Analogies. 



SOLUTION OF OBLIQUE SPHERICAL TRIANGLES. 131 

199. By examining the formulas [63] we reach the fol- 
lowing conclusions : 

1. In the first formula the factors cos -| (a — b) and cot ^ C are 
always positive; hence tan ^ (A -f- B) and cos |- (a -f- b) must always 
have the same sign. Therefore, if a-\- b < 180°, and consequently 
cos * (a + 6) > 0, then it follows that tan \ ( A -f- 5) > 0, and 
therefore i + £ < 180°. Similarly, it follows that if a -f 6 > 180°, 
then also A + B > 180°. . 

2. Also, if a + 6 = 180°, and consequently cos \ {a -f- b) = 0, then 
tan J (4 -f 5) = oo ; whence, \(A + B) = 90°, and i -f 5 = 180°. 

3. Conversely, it may be shown from the third formula that a'-\- b 
is less than, greater than, or equal to 180°, according as A -f- B is 
less than, greater than, or equal to 180°. 

Solution of Oblique Spherical Triangles. 
200. In the solution of oblique spherical triangles, 
there are six cases, as follows. Given, 

1. Two sides and their included angle. 

2. Two angles and their included side. 

3. Two sides and an angle opposite to one of them. 

4. Two angles and a side opposite to one of them. 

5. The three sides. 6. Tne three angles. 

Case I. 
201. Given two sides, a and b, and the included 
angle C. 

Method.— We find the angles A and B by the first and 
second of Napier's Analogies, viz. : 

fan* (A + B)~ COS f^T^ cotia 

2 v J cos \ (a + b) 2 

, . . A D . sin \(a — b) , , n 

tan Hi — B) = - — r~ — -— cot A C. 

2 y J sin \ (a -f b) 2 



132 



SPHERICAL TRIGONOMETRY. 



The side c may then be found by [53], or by the third 
or fourth of Napier's Analogies. It is better, however, to 
find c from one of Gauss's Equations, since they involve 
functions of the same angles that are used in the two 
formulas of Napier's Analogies. We can use any one 
of the formulas ; thus from the second we have 



cos j c 



COS -§-(« + &) • j n 

cosi(^+-B) Sm ^ C - 



EXERCISES XLII. 

1. In a spherical triangle, given a = 72° 36', b — 40° 44', 
and C= 54° 40'; find the other parts. 



Solution.— a = 72° 36'. 
b = 40° 44'. 
C=54 o# 40 / . 

log cos \ (a — b) = 9.982986 
colog cos \ (a + b) = 0.260025 

log cot I C = 10.286614 

log tan \ (A+B) =10.529625 
I (A+B) = 73° 32 / 39" 

log cos £ (a + b) = 9.739975 
colog cos i (A+B) = 0.547790 

lojrsin A O = 9.661970 



hence, 



i(a_ 6) = 15°56 / . 

£ (a + 6) = 56° 40'. 



;7° 20 y 



±C 



= 9.949735 
= 27° 02 / 16 A 



log sin £ (a — &) = 9.438572 

colog sin £ (a + b) = 0.078060 

log cot i C = 10.286614 

log tan £ (J— £) = 9.803246 

£ (^ _ .£) = 32° 26 / 37" 

%(A + B)= 73°32 / 39 // 

A =105° 59 / 16" 

B = 41° 06' 02" 

c = 54° 04 / 32" 



2. Given a = 80° 32' 40", & = 120° 27' 18", C= 48° 12' 
21"; find ^ = 57° 9' 4", 5 = 132° 45' 46", C = 61° 5' 4". 

3. Given a = 124° 50' 48", c = 75° 35' 50", 5 = 56° 36' 



26' 



find ^4=134° 10' 34", (7=57° 49' 36", b 



49' 18". 



SOLUTION OF OBLIQUE SPHERICAL TRIANGLES. 133 



Case II. 

202. Given two angles, A and B, and the in- 
cluded side c. 

Method. — We find the sides a and b by the third and 
fourth of Napier's Analogies : 



tan \ (a + b) — 



tan \ (a — b) 



tan i- c. 



cos J- (A — B) 
cos \ (A + B) 

sini(^ + £) tan2 



The angle C may then be found by the first or second 
of Napier's Analogies, or by one of Gauss's Equations. 
Thus, the first gives 

sin HA + £) 



cos 



2 ° 



cos \{a — b) 



cos £ c. 



EXERCISES XLIII. 

1. In a spherical triangle, given A = 108° 36' 45", .£ = 
40° 38' 28", c = 56° 42' 22" ; find the other parts. 



Solution.— J. = 108° 36 / 45". 


i(A — 5) = 33° 59' 8 J". 


B= 40°38 / 28 // . 


H^ + £) = 74°37 / 36|". 


c = 56° 42 / 22". 


£c = 28°21Ml". 


log cos | (A—B) = 9.918647 


log sin I (A—B) = 9.747401 


colog cos § (A+B) = 0.576582 


colog sin J (A+B) = 0.015824 


log tan \c = 9.732103 


log tan i c = 9.732103 


logtanHa+ h) =10.227332 


log tan ±(a — b) = 9.495328 


£(« + &) = 59° 2F 16" 


£(« — &) = 17° 22' 19" 


log sin I (A+B) = 9.984176 


I (a + b) = 59° 21' 16" 


colog cos | (a — 6) = 0.020276 


a = 76° 43' 35" 


' log cos I c = 9.944501 


b =41° 58' 57" 


log cos \ C = 9.948953 


C = 54° 28 / 40" 


I C = 27° 14 / 20" 




2. Given A = 130° 27' 38" 


, B = 110° 43' 20", c = 124 ( 



134 SPHERICAL TRIGONOMETRY. 

26' 37"; find a = 125° 55' 41", b = 84° 30' 55", 0= 129° 
12' 22". 

3. Given B = 148° 24' 36", C= 86° 38' 42", a = 88° 30' 
47" ; find b = 148° 21' 3", c = 89° 30' 25", A = 86° 21' 50". 

Case III. 

203. Given two sides a and b, and the angle A 
opposite one of them. 
Method. — The angle B is found from [53], from which 

we have 

. _ sin b sin A 

sin B = : • 

sin a 

Then C and c may be found from the fourth and second 
of Napier's Analogies, which give 

tan 4- c — — — 7-7--; ^r tan \ (a — b). 

z sin \{A — B) L ' 

cotja=^f^t|itanK^-£). 

2 sin £ (a — &) 2 

Note. — In this case, since B is found from the sine, there will 
sometimes be two solutions. If it is seen in the problem that 
B < 90°, there is but one solution. If in the calculation we find 
sin B > 1, the problem is impossible. 

The following truths may be readily deduced : 

1st. When A = 90°, there is only one solution, and may be no 
solution. 

2d. When A <C 90°, there are two solutions when a-\- b <d 180°, 
and a <C 6. 

3d. When A > 90°, there are two solutions when a-\- b >180°, 
and a^> b. 



SOLUTION OF OBLIQUE SPHERICAL TRIANGLES. 135 



EXERCISES XLIV. 

1. Given a = 53° 25', h = 34° 26', and A = 106° 35' ; find 
B, c, and C. 

Solution. — In this problem 

we have A > 90°, 

and a -f- b < 180°, 

hence, A + B < 180° ; 
whence, 5 < 90°, 

and there is only one solution. 
a-\-b= 87° 51' 
a — &= 18° 59' 
.4 + £ = 149° OF 43" 
^4 _ 5 = 64° 08' 17" 
log sin i (A + B) = 9.983941 
log tan J (a — 6) = 9.223218 
colog sin' I {A — B) = 0.274954 
log tan J c = 9.482113 

J c = 16° 52' 53" 
c = 33° 45' 46" 



log sin A (106° 35') = 9.981549 
log sin b ( 34° 36') = 9.752392 
colog sin a ( 53° 25') = 0.095289 
log sin B =9.829230 

B = 42° 26' 43" 
\ (a -f b) = 43° 55 / 30" 
J (a — 6)= 9° 29' 30" 
| (.4+5) = 74° 30' 51 J" 
J (.4— £) = 32°04 / 08J" 
log sin $(a + b)= 9.841181 
log tan i (A— B)= 9.796953 
colog sin i (a — 6) = 0.782768 
log cot $ (7 =10.420902 

J (7 = 20° 46' 36" 
(7 = 41° 33' 12" 



2. Given a = 75° 27' 40", b = 118° 45' 36", A = 84° 52' 
34"; find B = 115° 34' 27", c = 111 45' 16", (7= 107° 
07' 24". 

3. Given b = 40° 16', c = 47° 44', B = 52° 30' ; find 



C=65° 16' 35' 
C= 114° 43' 25' 



53° 19' 20", A = 79° 52' 22" ; or, 



— 17° .W 



4. Given a = 40° 20', b = 60° 30', and A = 50° 45'; show 
that the solution is impossible. 



136 SPHERICAL TRIGONOMETRY. 

Case IV. 

204. Given two angles, A and B, and the side 
a opposite one of them. 

Method. — The side a is found from [53], from which 
we have 

sin a sin B 



sin o = 



sin A 



Then c and C may be found from the fourth and second 
of Napier's Analogies, which give 

, , sin i (A ~\- B) , t , _ N 

tan * c= S miu-i) tan * (a - 6) - 

cot|C= sin f (a+ -^tanH^-i?)- 

2 sin £ {a — b) 2 v y 

Note. — In this case, since b is found from the sine, there will 
sometimes be two solutions, and may he no solution. If it is seen 
in the problem that b < 90°, there will be but one solution. If in 
the calculation we find sin b >• 1, there will be no solution. 

The following truths may be readily deduced : 

1st. When a = 90°, there is only one solution, and may be no 
solution. 

2d. When a < 90°, there are two solutions when A -J- B <C 180°, 
and A < B. 

3d. When a > 90°, there are two solutions when A + B > ]80°, 
and A > B. 

EXERCISES XLV. 

1. Given A = 112° 50', B = 135° 25', a = 150° 36'; find 
o = 158° 2' 40", c = 30° 45' 26", C= 73° 46' 46". 

2. Given 4 = 114° 36' 40", B = 82° 27' 18", 5 = 86° 20' 
30"; find a = 113° 45' 44", c = 82° 7' 18", (7= 79° 44' 2". 



SOLUTION OF OBLIQUE SPHERICAL TRIANGLES. 137 

3. Given A = 132° 16', B = 139° 44', b = 127° 30' ; find 
a = 65° 16' 35", C= 165° 41' 38", c = 162° 20' 38" ; or, a = 
114° 43' 25", <7 = 126° 40' 40", c = 100° 7' 38". 

4. Given J. = 60° 30',. £ = 40° 24', & = 50° 36'; show 
that the solution is impossible. 

Case V. * 

205. Given the three sides, a, b, and c. 

Method. — The angles may be found by the Formulas 
[56] or [57] or [58]. The formulas for the tangent, 
however, are, generally, to be preferred. 

The formulas for the tangent may be put in a still more 
convenient form by making 

sin ( s — a) sin ( s — b) sin (s — c) 2 

— tan t. 
sin s 

and substituting this in each and reducing, by which we 

obtain 

tan ^ A — tan r -~ sin (s — a). } 

tan \ B = tan r -i- sin (s — b). > [65] 

tan J O == tan r -r sin (s — c). J 

Notes. — 1. When only one angle is to be found, use For's. [56], 
[57], or [58] ; when all three angles are required, use For's. [65]. 

2. No ambiguity can arise in this case, since the half angles must 
be less than 90°. 

EXERCISES XLVI. 

1. Given a = 60° 34' 20", b = 48° 45' 26", c = 76° 48' 53" ; 
find A, B, and C. 



L38 



HrilKllKlAL TRIGONOMETRY. 



Solution. — The solution by Formula [58] is as follows: 



a= 60° 34/ 20" 

b = 48° 45' 26" 

c = 7 0° 4S' r>:>," 

2s =180° 08' 39" 



« = 32° 29' 59^ 



s — c= 16° 15' 26£" 



log sin ( 6 >_&)=r 9.844229 

log sin(s — c) = 9.447084 

colog sin (s — «) = 0.269786 

COlog sin .s- = Q.( )()()( >25 

2)19.561724 

log tan M= 9.780862 

\A = 31° 07' 18" 

.4 = 62° 14' 36" 



The angles /i and (7 are found in a similar manner. 

Solving the same problem by the three formulas [58], we have 



log tan I A= 9.780302 
log tan \ B = 9.666847 



£ .4 = 31° 07' 18" 
\ B = 24° 54' 28" 
|C = 49° 1 2' 22" 

,4 = 62° 14' 36" 
£==49°48',56" 
(7 =98° 24' 44" 



log sin (s — a) = 9.730214 
log sin (5 — &)= 9.844229 
log sin (s — c) = 9.447084 
colog sin s = O000625 

log tan 2 ?- =19.022152 
log tan r= 9.511076 
Note. — To find log tan $ J, we 
need not rewrite log tan r and log 
sin (s — a), but can subtract log sin 
s — a from log tan r as they stand 
in the first column ; and similarly 
for tan J />', and tan j> C. 

2. Given a = 120° 45' 28", ft = G2° 27' 40", c = 108° 23' 
40"; find .4 = 115° 44' 52", 13= 0S° 20' 24", C= 95° 
57' 32". 

3. Given a = 135° 16' 40'', & = 110° 55' 30", c = 86° 32' 
16"; find ^4 = 137° 38' 32", J5 = 116° 34' 34", (7= 107° 
06' 36". 

4. Given a = 25° 24' 23", b = 48° 38' 28", c = 76° 46' 
3G"; show that this is impossible. 



SOL UTION OF OBL IQ UE SPIIER TOA L TRIA NGLES. 139 



Case VI. 
20G. Given the three angles, A, B, and C. 
Method. — The sides may be found by the Formulas 
[59], [00]. or [61]. The formulas for the tangent are 
usually preferred, since they require fewer logarithms and 
give accurate results in every part of the quadrant. 

These formulas for the tangent may be put in a still 
more convenient form by substituting in each 

tan' 2 R = — cos S sec (S — A) sec (S — B) see (S — 0), 
which, when reduced, gives us 

tan \ a = tan R -f- sec (S — A). "] 

tan \ b = tan R - sec (S— B). I [G6] 

tan l c = tan R -r- sec (S — C). 

Note. — "When only one side is required, use For's. [59, 60, 61] j 
when all the sides are required, use For's. [60]. 

EXERCISES XLVIL 

10G° 36', B = 87° 45', C= 96° 48' ; find a, 



1. Given A 
b, and c. 

Solution. — Since the three sides 
are required, we solve by Form- 
ulas [66]. 

A = 106° 36' 
7i= 87° 45' 
C= 96° 4S / 
2 £=291° 09' 
log cos S= 9.916384 (n) 
log sec (S—A) = 10.109344 , 
ldg sec ($— B) = 10.273674 
log sec {S—C) = 10.181103 
log tan 2 R == 40.480505 
log tan R = 20.240252 



S—A-. 
S—B. 



145° 34 / 30" 
38° 58 / 30" 
57° 49' 30" 



S—C= 48° 46' 30" 
log tan £ a = 10.130908 
log tan J b = 9.966578 
log tan I c = 10.059149 

\a= 53° 30' 26" 

f b = 42° 47 / 51" 

" 2 1 c = 48°53 / 23 // 

a = 107° 00 / 52" 

b = 85°35 / 42" 

c= 97° 46' 46" 



140 



SPHERICAL TRIGONOMETRY. 



Note. — To find log tan | a, we need not rewrite log tan R and log 
sec (S — A), but can subtract them as they stand in the first column ; 
and similarly for log tan J b and log tan J c. 

2. Given A = 130° 46', B= 110° 50', C = 80° 30'; find 
a = 140° 32' 18", b = 128° 20' 40", c = 55° 51' 28". 

3. Given A = 60° 25' 40", B = 87° 26' 32", C = 60° 25' 
40" ; find a = 53° 36' 16", b = 67° 36' 20", c = 53° 36' 
16". 

4. Given A = 90°, B = 90°, C= 90° ; find a = 90°, & = 
90°, and c = 90°. 



Solution by Means of a Perpendicular. 

207. Oblique spherical triangles may be readily solved, 
also, by dividing them into right triangles and applying 
Napier's Rules. 

Thus, let CD be a perpendicular 
drawn from C to the base AB. 

1. Then, Rule II. 

cos a = cos m cos h. 
Whence, cos h = cos a -j- cos m. 

* cos b = cos % cos h. 

Whence, cos h = cos b -+- cos n. 

Whence, cos a : cos m= cos& : cos n. 

That is, £/ie cosines of the sides are proportional to the cosines 
of the segments of the base. 

2. Again, by Rule II. 

cos A = cosh sin A CD. 

cos B — cos h sin BCD. 

Whence, cos A : cos B — sin A CD : sin BCD. 




SOLUTION BY MEANS OF A PERPENDICULAR. 141 

That is, the cosines of the angles at the base are proportional 
to the sines of the segments of the vertical angle. 

3. Again, by Rule I. 

sin n = tan h cot A — tan h -f- tan A. 
sin m — tan h cot B = tan h -f- tan B. 
Whence, sin m : sin n = tan A : tan B. 

That is, the sines of the segments of the base are inversely 
'proportional to the tangents of the angles at the base. 

4. Again, by Rule I. 

cos ACD = tan h cot b. 
cos BCD = tan h cot a. 
Whence, cot a : cot b — cos BCD : cos ACD. 
That is, the cotangents of the two sides are proportional to the 
cosines of the segments of the vertical angle. 

5. Again, by Art. 207, we have 

cos a : cos b = cos m : cos n. 
Whence, 

cos &+cos a . cos &_ cos a=cos n-fcos m : cos n— cos m. 
But, Art. 70, 

cos &+cos a : cos b — cos a=cot ^(a+b) : tan \ (a — b). 

cos 71+cos m : cos n— cos m=cot \ (m+n) : tan J (m — ?i). 
Whence, 

cot J (a+6) : cot J (m+w)=tan -J- (a — &) : tan \ (m — n). 

And since tangents are inversely proportioned to cotan- 
gents, 

cot \ (a+&) : cot \ (m+n)=tan \ (ra-fn) : tan J (a+J). 

Whence, 

tan \ (ra-j-n) : tan -J (<x-f &)=tan| (a — &) : tan \ (m — n). 



142 



SPHERICAL TRIGONOMETRY. 



That is, the tangent of half the sum of the segments of the base 
is to the tangent of half the sum of the sides, as the tangent of 
half the difference of the sides is to the tangent of half the 
difference of the segments of the base. 

208. These five principles, derived immediately from 
Napier's Rules, enable us to solve every case of the oblique 
triangle. They are more easily remembered than the 
formulas previously used, and are preferred by some 
authors in solving these triangles. 



EXERCISES XLVIII. 

1. In the spherical triangle ABC, given AC= 70° 30', 
5(7=80° 36', and the angle ^4 = 35° 24'; required the 
other parts. 

Solution. — Let ABC denote the triangle. C 

Draw CD J_ to AB. 

First, we have, Art. 183, 

sin BC : sin A C = sin A : sin B. 
Whence, B = 33° 36' 23". 

Then in triangle ACT), Rule I., 
cos AC = cot A cot A CD. 
Whence, ACD = 76° 39" 17". 

Also in triangle BCD, 

cos BC= cot B cot BCD. 




Whence, 



BCD= 83° 48' 19". 
.4(75 = 160° 27' 36". 



Fig. 58. 



Therefore, 

Finally, we have 

sin A : sin (7= sin BC : sin AB. 
Whence, AB = 145° 16' 33". 



SOLUTION BY MEANS OF A PERPENDICULAR. 143 

2. In the spherical triangle ABC, given A = 114° 36' 
40", B = 82° 27' 18", and AC = 86° 20' 30"; find BC = 
113° 45' 44", 45 = 82° 7' 18", and ACB = 79° 44' 3". 



3. In the spherical triangle ABC, given 4£ = 72° 36', 
AC= 40° 44', and 4 = 54° 40' ; find B = 41° 06' 02", C = 
105° 59' 16", and BC= 54° 04' 32". 

4. In the spherical triangle ABC, given A = 108° 36' 
45", (7= 40° 38' 28", and AC = 56° 42' 22"; find 5C = 
76° 43' 35", B = 54° 28' 40", and AB = 41° 58' 57". 

5. In the spherical triangle ABC, given AB = 112° 25', 
4C= 60° 20', BC*= 81° 10' ; find A = 64° 46' 36", B = 
52° 42' 12", and (7= 122° 11' 06"! 

6. In the spherical triangle ABC, given A = 106° 36', 
^ = 87° 45', C= 96° 48' ; find AB = 97° 46' 46", BC = 
107° 00' 55", 4C = 85° 35' 42". 

Notes. — The following suggestions will aid the student with the 
above examples. 

1. In Ex. i, since the value of AB is found from the sine, we 
determine its quadrant by Art. 160. 

2. In Ex. 2, we first find BC by Art. 184; then by Rule I. find 
AD and BD ; then take their sum and find ACB by Art. 184. 

3. In Ex. 3, we first find AD by Rule I. ; then find B by For. 3, 
Art. 207; then BC by Rule I., then BCD by Rule I., and then 
A CD by Rule I., from which we find C. The latter part of this 
solution prevents ambiguity. 

4. In Ex. 4, first find ACD by Rule L, from which find BCD; 
then find BC by For. 4, Art. 207 ; then find B by Rule I. ; then find 
AB by Art. 184. 

5. In Ex. 5, first find AD and BD by Art. 184 ; then find A by Rule 
I. ; then find B by Rule I. ; then find C by Art. 188, 

6. In Ex. 6, pass to the polar triangle; find its angles as in Ex. 5 ; 
the supplements of these angles will be the sides of the given 
triangle. 



144 SPHERICAL TRIGONOMETRY. 



SECTION XV. 

SUPPLEMENT. 

Area of a Spherical Triangle. 

209. We now proceed to show how to find the area of a spherical 
triangle. 

I. When the three angles, A, B, and C, are given. 

Let R = the radius of the sphere. 

E — the spherical excess = A -f B-\- C— 180°. 

S = the area of 'the triangle. 
Then by Geometry, B. IX., Th. XXVII., 

S = 180. ' 

II. When the three sides are given. 

210. Take E= A + B+ (7—180° as above ; then 

tan IE=^^ A -hB+C-lSO) 
cos I (A + B + C— 180) 

= sin $(A + B) — sin j (180 — C) 
cos I (.4 + B) + cos J (180 — C) 

Art. 70. 
_ sin | (J 4- B) — cos £ C 
cos I (J. 4- -6) 4- sin £ C 
_ cos j (a — b) — cos |- c cos ^ C' _ 
cos J {a -\- b) 4 cos J c sin -| 6 T 

For. [62] 

sin \ (c-\-a — b) sin \(c-{-b — a) I ( sin s sin (.s — e) | 

sin I (a-{-b-\-c) cos £ (a-\-b — c) \ ( sin (s — a) sin (s — b) j 

For. [56, 57]. 

_ sin \ (s — b) sin \ (s — a) I ( sin s sin (.9 — c) | , 

cos £ s cos 2" (•? — c) \ ( sin (s — a) sin (5 — b) ) 



CIRCUMSCRIBED AND INSCRIBED CIRCLES. 145 



T,/sin s sin£(s — b) j/sin (g — c) sin \ (s — a) 
cos £ s j/sin (s — b) cos ^ (s — c) j/sin (s — a) 

Substituting for cos %s, sin |^ (s — b), etc., their values of [26] 
and [25], and reducing, we have 

tan \ E=-j/{tan \ s tan \ (s — a) tan \ (s — b) tan h (s — c). [67] 

211. This elegant formula is known as L'Huiller's Theorem. By 
means of it the value of E may be found from the three sides, and 
then the area of the triangle may be found from Art. 209; 

212. In a similar manner we can find Cagnoli's Theorem, 
which is 

. x t/ {sin s sin (s — a) sin (s — b) sin (s — c)} 

2 cos \ a cos \ b cos \ c 



Circumscribed and Inscribed Circles. 
I. To find the radial arc of a circumscribed circle. 

213. Let be the pole of the small 
circle circumscribed about the spherical 
triangle ABC. Draw the radial arcs 
OA, OB, and OC, and draw OD per- 
pendicular to BC. The triangles OB C, 
AOC, and A OB are isosceles, and BD = 
^ a. Denote the "angles by A, B, C. 

Denote the radial arc of the circum- 
scribed circle by R. 

Then in the right triangle BOD 




Fig. 59. 



we have 


cos OBD = cot R tan %a, 




Whence, 


r, tan ha 

tan R = 

cos OBD 




Now, 


OBD = B—ABO = B — BAO. 




And, 


OBD= OCD= C—ACQ= C- 


-OAC. 



10 



146 



SPHERIC A L TRIG ONOMETR Y. 



Whence, 2 OBI) = B-\-C — (BAO + OA C) 
= B + C — A = 2(S—A), 
hence, OBD = S— A. • 

tan ^ a 



Whence, tan R = 
Similarly, tan R 
Whence, tan 3 R 



cos(S-A) 

tan h b , , „ tan h c 

— - — — , and tan R = — - — — • 

cos (S—B)' cos (S—C) 

tan j a tan J b tan -| c 



cos (S—A) cos (S — B) cos (S— C) 
The product of the three formulas, [61], gives 
tan 2 £ a tan 2 \ b tan 2 J c = — 



3 3 S 



cos (S — A) cos (S — B) cos (S 
Substituting this in the value of tan 3 R and reducing 



tan R 



COS aS 



cos (S — A) cos (S — B) cos (S — C) 



C) 



[68] 



II. To find the radial arc of an inscribed circle. 

214. Let O be the pole of the circle in- 
scribed in the spherical triangle ABC; 
Draw the radial arcs OD, OE, and OF 
perpendicular to BC, AC, and AD respect- 
ively. Draw also the arc EGF. 

Now, since OE = OF, the triangle EOF B 
is isosceles. Then 
Z OFG = Z OEG and Z GFA =Z GEA. 

Hence, A FAE is isosceles ; and AF= AE. 

DraAV the arc OG perpendicular to EF at its middle point; it will 
bisect the angle EOF, and will also pass through the point A and 
bisect the angle A. Similarly, the arcs OB and OC will bisect the 
angles B and C respectively. Denote the radial arc of the inscribed 
circle by r. 




CIRCUMSCRIBED AND INSCRIBED CIRCLES. 147 

Then in the right triangle A OF # 

sin AF= tan r cot \ A. 
Now, AF=AE;BF=BD-, CE=CD. 

'Also, AF=c — BF=c — BD. 

And, AE = b — CE = b — CD. 

Adding, XF+ X£= 2 ^li^ 6 + c — (BD + CD). 
Or, 2AF=b+c— a. 

= 2{s-a). 
AF=s — a. 

Substituting this value in sin AF and reducing, 
We have tan r = sin (s — a) tan \ A. 

Similarly, tan r = sin (s — b) tan § B. 

tan r = sin (5 — c) tan ^ C. 

The product of these three formulas gives 

tan 3 r = sin (s — a) sin (s — b) sin (s — c) tan | J. tan -| J3 tan ^ C. 

Finding the value of tan % A tan ^ B tan | C from For's. [58], 
and substituting and reducing, we have 



tan 



r = J sin ( g — g) sin ( g — b ) sin ( g ~ g ) I [69] 

\ sin 5 i 



EXERCISES XLIX. 

1. Find the area of a spherical triangle whose angles are 60° 30 / , 
70° 40', and 80° 50'. 

2. Find the area of a spherical triangle whose sides are 60° 30', 
70° 40 / and 80° 50'. 

3. Find the radius of the circumscribed circle in the triangle of 
Exercise 1. 

4. Find the radius of the inscribed circle in the triangle of 
Exercise 2. 



148 SPHERICAL TRIGONOMETRY. 

Miscellaneous Exercises. 

1. In any spherical triangle, if A = a, show that B = b, and 
C=c, or that they are respectively supplementary. 

2. When does the polar triangle coincide with the primitive 
triangle? 

3. If B = A + and D is the middle point of b, prove that b = 
2 AD. 

4. If D is the middle point of c, prove that 

cos b -f- cos a = 2 cos £ c cos CD. 

5. If 6 -j- c = 7r, prove that sin 25-f- sin 2 C=0. 

6. In an equilateral spherical triangle, prove that 2 cos \ a sin \ 
A = l. 

7. In an equilateral spherical triangle, prove that tan 2 ^ a = 
1 — 2 cos A. 

8. In an equilateral spherical triangle, prove that sec A = l-\- 
sec a. 

9. If b = c = 2a, proye that esc \ A = 4 cos a cos J a. 

Right Spherical Triangle. 

1. Derive two rules similar to those of Napier for the direct solu- 
tion of quadrantal triangles. 

If ABC is a right triangle, C being the right angle, then 

2. Prove sin 2 -| c — sin 2 \ a cos 2 \b -\- cos 2 \ a sin 2 \ b. 

3. Prove tan \ (c -{- a) tan £ (c — a) = tan 2 \ b. 
1 4. Prove sin (c — 6) = tan 2 -| J. sin (c + &). 

5. Prove sin a tan |- J. — sin b tan ^ i? = sin (a — b). 

6. Prove sin (c — a) = sin b cos a tan J B. 

sin (c — a) = tan b cos c tan \ B. 



OBLIQUE SPHERICAL TRIANGLES. 149 

7. Prove sin (c -f- a) = sin b cos a cot \ B. 

sin (c -j- a) = tan 6 cos c cot £ ^« 

8. In a right spherical triangle, C the right angle, if D is the 
middle point of AB, prove that 

sin 2 a -j- sin 2 6=4 cos 2 J c sin 2 CD. 

9. In a right triangle, if d is the length of the arc from C perpen- 
dicular to the hypotenuse, prove that 

cot 2 a -f- cot 2 b = cot 2 cl. 

10. If ABC is a right spherical triangle, A not being the right 
angle, prove that if A == a, then b and c are quadrants. 

In a right triangle, C being the right angle, prove 

11. tan' i A = "j" <—»> ■■ 13. tan*(45°-M= to * ("^ 

sin (c -{- 6) tan £ (c -}- «) 

12. tan 2 ^c = -^M+^). 14 cos £ = sin2^ < 

cos (A — B) cos 6- sin 2 /? 

Oblique Spherical Triangles. 

1. If the area of an equilateral triangle is one-fourth of the area 
of the sphere, what are its sides and angles ? 

2. In a spherical triangle, if c = 90° and d denotes the perpendic- 
ular from C to c, then cos 2 d = cos 2 a -f- cos 2 b. 

3. In a spherical triangle, if A = B = 2 C ; then 

8 sin (a -f- \ c) sin 2 \ c cos \ c — sin 3 a. 

4. In a spherical triangle, if A — i? = 2 C ; then 

8 sin 2 I C (cos s -J— sin ^ C) cos £ c — cos #. 

5. In any equilateral triangle, R and r denoting respectively the 
radii of circumscribed and inscribed circles, prove tan R = 2 tan r. 

6. In any spherical triangle, E denoting the spherical excess, 

prove 

sin ^ E = sin C sin J a sin | 6 sec \ c. 



150 SPHERICAL TRIGONOMETRY. 

7. In any spherical triangle, E denoting the spherical excess, 
prove 

cos % E = { cos £ a cos £ b -j- sin h a sin J 6 cos C} sec \ c. 

8. If the angle C of a spherical triangle is a right angle, prove 

sin J E== sin £ a sin £ 6 sec £ c ; 
cos % E= cos J a cos f 6 sec J c\ 

9. If the angle C is a right angle, prove that 

sin 2 c -„ sin 2 a , sin 2 b 

cos E = 1 • 

cos c cos a cos 6 

10. If a = b and C .= — ', prove that E = « 

2 2 cos a 

11. If the angles of a spherical triangle are together equal to four 
right angles, prove 

cos 2 ^ a -\- cos 2 \ b ~\- cos 2 fc= 1. 

12. If ABC is an equilateral spherical triangle, P the pole of the 
circumscribed circle, and Q any point on the sphere, prove that 

cos AQ A- cos BQA- cos CQ = 3 cos R cos PQ. 

13. Find the surface of an equilateral and equiangular spherical 
polygon of n sides, and determine the value of each of the angles 
when the surface equals one-half the surface of the sphere. 



A TABLE 



OF 



LOGARITHMS OF NUMBERS 

From 1 to 10,000. 



N. 


Log. 


N. 


Log. 


N. 


Log. 


N. 


Log. 


, 


0- 000000 


26 


I-4I4973 


5i 


1-707570 


76 


r -880814 


2 


o-3oio3o 


27 


1 -43i J64 


52 


1 -716003 


77 


1-886491 


3 


0-477121 


28 


1 -447108 


53 


1-724276 


78 


1-892085 


4 


o- 602060 


29 


i-4'>2398 


54 


1 --32394 


79 


1 -897627 


5 


0-698970 


3o 


1 -4771 2I 


55 


1 -74o363 


80 


1 -9o3oqo 


6 


0-77801 


3i 


1 -491062 


56 


1 -748188 


81 


1-908485 


i 


0-843098 


32 


1 -5o5i5o 


57 


1 -755873 


82 


i- 9 i38i4 


0-903090 


33 


1 -5i85i4 


58 


1-763428 


83 


1 -919078 


9 


0-954243 


34 


1-53U79 


5 9 


1 '770S52 


84 


1-924279 


10 


I -000000 


35 


1.544068 


60 


1. 778101 


80 


1 -929410 


n 


I -041393 


36 


1 -5563o3 


61 


i-78533o 


86 


1 -934498 


12 


I -079181 


37 


1-568202 


62 


1-792392 


87 


1 -939019 


i3 


I-I 13943 


38 


1 -079784 


63 


1 -799341 


83 


1-944483 


U 


1-146128 


3 9 


1 -59io65 


64 


1-806181 


89 


1 .949390 


i5 


I • 1 76091 


40 


1 -602060 


65 


1 -812913 


90 


1 -954243 


16 


I • 2041 20 


41 


1 -612784 


66 


I- 819644 


9 1 


1 -909041 


H 


I • 230449 


42 


1 -623249 


67 


1 -826070 


92 


1-963788 


1 8 


I -255273 


43 


1-633468 


68 


1 -83 2 509 


93 


1-968483 


19 


I-278754 


44 


1-643453 


69 


1-838849 


94 


1 -973128 


20 


i -3oio3o 


45 


1 -6532i3 


70 


1-845098 


9 5 


1-977724 


21 


1 -322219 


46 


1-662753 


7i 


1-851258 


96 


1 -982271 


2 2 


1-342423 


47 


1 -672098 


72 


1-857333 


97 


1-986772 


23 


1-361728 


48 


1 -681241 


73 


1-863323 


98. 


1 -991226 


24 


i-3So2ii 


49 


1 -690196 


74 


1-869232 


99 


1 -Q95635 


25 


1-397940 


5o 


1 -698970 


V 


1 -870061 


100 


2 -000000 



Eemark. — In the following table, in the nine right- 
hand columns of each page, where the first or lead- 
ing figures change from 9's to O's, points or dots are 
introduced instead of the O's, to catch the eye, and to 
indicate that from thence the two figures of the Log- 
arithm to be taken from the second column, stand in 
the next line below. 



A TABLE OF LOGARITHMS FROM 1 TO 10,000. 



N. 

IOO 





1 


2 | 3 | 4 | 5 | 6 | 7 | 8 


9 


L>. 


000000 


0434 


0868 i3oi 1734 1 2166 1 2598: 3029 


346i 


38 9 i| 432 


101 


432i 


475i 


5i8i 56o 9 | 6o38 


6466 1 6894, 7321 


7748 


8174I 428 


I02 


8600 


9026 


945 1 9876 »3oo 


•724! 1 147 1370 1993 


24i5; 424 


io3 


012837 


J2D9 368o| 4ioo| 452i 


4940J 536o' 5779' 6197 


6616 419 


1 04 


7o33 


745 1 


7868: 82841 8700 


0116 9532; 9947" " *36i 


• 77 5 


416 


io5 


02 1 1 89 


i6o3 


2016J 2428I 2841 


3252 


3664' 4073! 4486 


4896 


412 


106 


53o6 


570 


6i25 6533 1 6o42 


735o 


7-^7! 8164 8571 


8978 


408 


107 


o384 
o33424 


9789 
3826 


•igSi «'6oo 1004 


1408 


1812, 221b 2619 


3o2i 


404 


108 


4227 4628 5029 


543o 


583o 6?3o 6629 


7028 


400 


105 


7426 


7825 


8223; 8620I 901-" 


9414 
3362 


9R1I °207j ®6o2 


•998 


3g6 


1 10 


041393 


1787 


2182 2576 2g6g 


3755 4<48i 4340 


4g32 


3 9 3 

38g 


1 1 1 


5323 


5714 


6io5 


6495 6885 


7273 


7664! 8o53 


8442 


883o 


113 


9218 

053078 


9606 


0993 


•38o\ '766 


Si 53 


1 538! 1924 


2309 


2694 


386 


n3 


3463 


3846 


423o 


46i3 


4996 


53 7 8 5760 


6142 


6324 


382 


H4 


6905 


7286 


7666 


8046 


8426 


88o5 


9 1 85: o563 
2958J 3333 


9942 
3709 


°320 


379 


1 1 5 


060698 


1075 


1432 


1829 


2206 


2382 


4o83 


376 


116 


4458 


4832 


52o6! 558o 


5953 


6326 


6699; .7071 
0407! * 77 6 


7443 


78i5 


3 7 2 


117 


8186 


8557 


8928 


9298 


9668 


••38 


1143 


i5i4 


36g 


118 


071882 


225o 


2617 


2985 


3352 


3 7 i8 


4o85i 445i 


4816 


5i82 


366 


119 


5547 


5912 

o543 
3i44 


6276 


6640 


7004 


7368 


773i 8094 


8457 


8819 


363 


120 


079181 


9904 
35o3 


•266 


•626 


•987 
45 7 6 


i347j 1707 


2067 


2426 


36o 


121 


0S2785 


386i 


4219 


4g34i 52 9 i 


5647 


6004 


35 7 
355 


122 


636o i 6716 


7071 


7426 


7781 


8i36 


8490I 8845 


9198 


g552 
3071 


123 


9905 


•258 


•611 


♦963 


i3i5 


1667 


2018I 2370 


2721 


35i 


124 


093422 


3772 


4122 


4471 


4820 


5i6 9 


55i8| 5866 


621D 


6562 


349 


125 


6910 


7 25 7 


7604 


7 9 5 1 


8298 


8644 


8990J 9335 


9681 
3ng 


°»26 


346 


126 


100371 


0715 


io5 9 


i4o3 


1747 


2091 


24341 2777 


3462 


343 


127 


3So4 


4146 


4487 


4828 


5i6 9 


55io 


585 1 1 6191 


653 1 


6871 


340 


128 


7210 


7549 


7888 


8227 


8563 


8903 


9241 9 5 79 


9916 


•253 


338 


129 


1 1 oSgo 


0926 


1263 


i5 99 


1934 


2270 


26o5 2940 


3273 


36og 


335 


i3o 


n3 9 43 


4277 


461 1 


4944 


52 7 8 


56 1 1 


5943! 6276 


6608 


6940 


333 


i3i 


7271 


7603 


/ 9 34 


8265 


85 9 5 


8926 


9256] g586 


99 1 5 
3i 9 8 


•243 


33o 


1 32 


120574 


0903 


123l 


i56o 


1888 


2216 


2544 


2871 


3523 


328 


i33 


3852 


4178 


45o4 


483o 


5i56 


5481 


58o6 


6i3i 


6456 


6781 


325 


i34 


7io5 


7429 


7753 


8076 


83 99 


8722 


90 4 5 


9368 


9690 


00 , 2 


323 


i35 


i3o334 


06 5 5 


0977 


1298 


i6iq 


i 9 3 9 


2260 


258o 


2900 


32ig 


321 


i36 


353q 


3858 4"? 


4496 


4814 


5.33 


545i 


5769 


6086 


6403 


3i8 


1 3 7 


6721 


7037 


7354 


7671 


7987 


83o3 


8618 


8 9 34 


9249 


9364 


3 1 5 


1 38 


9879 


•194 


•5o8 


•822 


n36 


i45o 


1763 


2076 


238 9 


2702 


3i4 


1 3 9 


i43oi5 


3327 


363 9 
674S 


3 9 5i 


4263 


4374 


4885 


5i 9 6 


55o7 


58 1 8 


3n 


140 


146128 


6438 


7o58 


7 36 7 


7676 


79 85 


8294 


"86o3 


891 1 j 3og 


141 


9219 


9527 


9835 


•142 


®44g 


•756 


io63j 1370 


1676 


1982 307 


142 


1522S8 


2D 9 4 


2900 


32o5 


35io 


38i5 


4120 4424 


4728 


5o32 3o5 


143 


5336 


5640 


5 9 43 


6246 


6549 


6852 


7 '34J 7437 


7739 


806 1| 3o3 


«44 


8362 


8664 


8960 


9266 


9 56 7 


9868 


•168; «46g 


•769 


10681 3oi 


145 


i6i363 


1667 


1967 


2266 


2564 


2863 


3i6i| 346o 


3758 


4o55| 299 


146 


4353 


465o 


4947 


5244 


554i 


5838 


6i34' 643o 


6726 


7022 297 


'47 


7317 


76i3 


7908 


8203 


8497 


8792 


9086 1, 9380 


9674 


9968I 293 


148 


170262 


o555 


0848 


.1141 


1434 


1726 


20.19 23 11 


26o3 


2895I 293 


149 


3 1 86 


3478 


3769 


4060 


43 5 1 


4641 


4932i 5222 


55 1 2 


58o2| 291 


I JO 


1 7609 1 


638i 


6670 


6959 


7248 


7536 


7825 81 i3 


8.',o 1 


8680; 289 


i5i 


8977 


92641 9^52 


9 si 9 


•126 


•4 1 3 


•699 »o85 


1272 


i55S 287 


I 52 


18184.4 


2129 24i5 


2700 


2 9 H5 


3270 


3555 3839 


4123 


4407 j 285 


1 53 


4691 


4975) 5 2 5 9 
7803 8084 


5542 5825 


6108 


6391 6674 


6g56 


723gi 283 


1 54 


7021 


8366 8647 


8928 


9209 9490 9771 


••5 1. 281 


1 55 


190332 


06121 0892 


1171; 1 45 1 


n3o 


2010 2289! 2567 


2846: 279 


156 


3 1 2 5 


34o3 368 1 


3g5g' 4237 


45i4 


4792 5o6g ; 5346 


5623' 278 


1 37 


58991 0! 7° 453 6729' 70*05 7281 


7556 7832 8107 


8382; 276 


1 58 


8657' 8932 9206 9481^ 9755 ®*29 


•3o3 »577! «85o 


1 1 24 274 


1 5 9 


201397I 16701 1943 2216 2488 2761' 3o33 33o5 3577 


3848! 272 


i N « 


| I I 2 | 3 | ■ 4 j 5 | 6 | 7 1 8 i 


T~ii5r 



A TABLE OF LOGARITHMS FROM 1 TO 10,000. 



N. 





1 


2 1 3 1 4 


5 


6 


7 


8 


9 I D. 


1 60 


204120 


4391 


4663 4g34 52o4 


5475 


5746 


6016 


6286 


6556 271 


161 


'6826 


7096 


7365 i 7634, 7904 


8i 7 3 


8441 


8/710 


8979 


9247I 269 


162 


95i5 


97 83 
2454 


••5i| *2>\q' «586 


•853 


1 121 


1388 


1654 


1 92 1 207 


1 63 


212188 


2720; 2986 3252 


35 18 


3783 


4049 


43 1 4 


4579 2C6 


164 


4844 


5109 


53 7 3 5638 


5902 


6166 


643o 


6694 


6957 


7221! 264 


i65 


7484 


7747 


8010 8273 


8536 


8798 


9060 


9323 


9585 


9846] 262 


166 


220108 


0370 


o63i 0892 


11 53 


1414 


1675 


1936 


2196 


2456 261 


167 


2716 


2976 


32361 3496 


3755 


4oi5 


4274 


4533 


4792 


5o5i 25g 


168 


5309 


5568 


58261 6084 


6342 


6600 


6858 


71 15 


7372 


763o; 253 


169 


7887 


8i44 


8.<oo| 8657 


8 9 i3 


9170 


9426 


9682 


9938 


•193 256 


170 


230449 


0704 


0960J 1 21 5 


1470 


1724 


1979 

45i7 


2 234 


2488 


2742 


254 


171 


2996 


325o 


3do4 3757 


401 1 


4264 


4770 


5o23 


5276 


253 


172 


3028 


5 7 8i 


6o33, 6285 


653 7 


6789 


7041 


7292 


7544 


77 9 5 


252 


i 7 3 


8046 


8297 


85481 8799 


9049 


9299 


95 5u 


9800 


••5o 


•3oo 25o 


174 


340549 


0799 


1048 1297 
3534 3782 


1546 


179D 


2044 


2293 


2541 


2790 249 


i 7 5 


3o38 


3286 


4o3o 


4277 


4523 


4772 


5019 


5266 248 


•176 


55i3 


5759 


6006 6252 


6499 


6745 


6991 


7237 


743 2 


7728 246 


177 


7973 


8219 


8464 8709 


8954 


9198 


9443 


9687 


9932 


•176 


245 


178 


250420 


0664 


0908 n5i 


i3g5 


1638 


IS8I 


2125 


2368 


2610 


243 


H9 


2853 


3096 


3338 358o 


3822 


4064 


43o6 


4548 


479° 


5o3i 


242 


180 


255273 


55i4 


5755 5 99 6 
8i53l 83 9 8 


6237 


6477 


6718 


6958 


7198 


7439 


241 


181 


7679 


7918 


8637 


8877 


9116 


9355 


9 5 9 4 


9833 


239 
238 


182 


260071 


o3io 


o548 


0787 


1025 


1263 


i5oi 


1739 


1976 


2214 


183 


245i 


2688 


2925 


3i62 


3399 


3636 


38 7 3 


4109 


4346 


4582 


23 7 


18.; 


4818 


5o54 


5290 


5525 


5761 


5 99 6 


6232 


6467 


6702 


6 9 3 7 


235 


iS5 


7172 


74o6 


7641 


•7875 


8110 


8344 


8578 


88l2 


9046 


9279 


234 


186 


9 di3 


9746 


99S0 


•2l3 


•446 


•679 


•912 


1U4 


1377 


1609 


233 


187 


271842 


2074 


23o6 


2538 


2770 


3ooi 


3233 


3464 


36 9 6 


3927 


232 


188 


4i58 


4389 


4620 


485o 


5o3i 


53 u 


5542 


5772 


6002 


6232 


23o 


189 


6462 


6692 


6921 


7 i5i 


738o 


7609 


7838 


8067 


8296 


8525 


220 


190 


278704 


8982 


92 1 1 


943o 
1 7 1 5 


9667 


9893 


•123 


•35i 


•578 


•806 


2 28 


191 


28io33 


1261 


1488 


1942 


2169 


2396 


2622 


2849 


3075 


227 


192 


33oi 


352 7 


3753 


3q"'9 


42o5 


443 1 


4656 


4832 


5107 


5332 


226 


193 


5557 


5782 


6007 


6232 


6456 


6681 


6905 


7 i3o 


7354 


7 5 7 8 


225 


194 


7802 


8026 


8249 


8473 


8696 


8920 


9.43 


9 366 


9 58 9 


9812 


223 


19 5 


290035 


0257 


0480 


0702 


092:") 


1147 


1369 


1 59 1 


i8i3 


2o34 


222 


19& 


2256 


2478 


2699 


2920 


3.41 


3363 


3534 


38o4 


4025 


4246 


231 


'97 


4466 


4687 


4907 
7104 


5*127 


5347 


•5567 
7761 


5787 


6007 


6226 


6446 


220 


198 


6665 


6884 


7 3 2 3 


7042 


7979 


8.9H 


8416 


8635 


219 


199 


8853 


9071 


9289 


9 5 °7 


9725 


9943 


•,6i 


•3 7 8 


•595 


•8i3 


2l8 


200 


3oio3o 


. 1 247 


1464 


1681 


1808 

403Q 


2 1 1 4 


2.33i 


2047 


2764 


2980 


217 


201 


3ig6 


3412 


3628 


3344 


4275 


4491 


4706 


4921 


5i36 


2l6 


202 


535i 


5566 


5 7 8i 


5906 ' 62 1 1 


6425 


663g 


6854 


7068 


72S2 


2 r 5 


203 


7496 


7710 


7924 


8137! 835i 


8564 


8778 


899' 


9204 


9417 


2l3 


?o4 


9630 


9343 


•°56 


©268| «48i 


•6 9 3 


•906 


1118 


i33o 


i542 


212 


2o5 


3in54 


1966 
4078 


2177 


2389 2600 


2812 


3o23 


3234 


3445 


3656 


211 


206 


3867 


4289 


4499 47io 


4920 


5i3o 


5340 


555i 


5760 


210 


207 


5 97 o 


6180 


6390 


6399 


6809 


7018 


7227 


7436 


7646 


7854 


2on 


208 


8o63 


8272 


8481 


8689 


8898 


9106 


9 3.4 


9322 


973o 


9 9 38 


208 


209- 


320146 


o354 


o562 


0769 


0977 


11 84 


i3gi 


i5 9 3 


i8o5 


2012 


207 


21 "> 


322219 


2^26 


2633 


283g 


3046 


3252 


3458 


366*5 


38 7 i 


40771 2(.6 


211 


4282 


4488 


4694 


4899 


5io5 


53io 


55i6 


5721 


5926 


6i3i 2o5 


V.l 2 


6336 


654i 


6745 


6900 


7 1 55 


7359 


7563 


7767 


7972 


8176J 204 


213 


838o 


8583 


8787 


8991 


9194 


9 3 9 8 


9601 


9805 


•®«8 


®2 I 1 i 2o3 


214 


33o4i4 


0617 


0819 
2842 


1022 


1225 


1427 


i63o 


1 83 2 


2o34 


2236 202 


2l5 


2438 


2640 


3o44 


3246 


3447 


3649 


3S5o 


4o5i 


4253 202 


216 


4454 


4655 


4856 


5o57 


52 57 


5458 


5658 


5S5o 


6o5g 


6260 201 


217 


6460 


6660 


6860 


7060 


7260 


7459 


7609 


7 S58 


8o58 


8257 200 


218 


8406 


8656 


8855 


oo54 


9253 


945 1 


9630 


98.-J9 


00 j-y 


•246 190 


-n 9 


340444 


cM-2 


oS4t' io3oJ 1237 


1415 


1 A3 2 


i83o 


2028 


2225 


198 


N. 





I 


2 13,4 


5 


6 


7 


8 


~ 


"a 



4 


A TABLE 


OF 


LOGARITHMS FROM 1 


TO 


10,000. 




N. 


| I | 2 


3 


4 


5 


6 


7 


8 j 9 


D. 


220 


342423! 2620 2817 


3oi4 


3212 


3409 


36o6 


38o2 


3999) 41961 197 
59621 6 1 D7J 196 


221 


4392 4389 


4780 


4981 


5i 7 8 


53 7 4 


5570 


5766 


222 


6353 j 6549 


6744 


6 9 3 9 


7i35 


733o 


7523 


7720 


79x5j 8110 195 


223 


83o5> 85oo 


8694 


8889 


9083 


9278 


9472 


966C 


9860; ®®54! 194 


2"4 


350243 


0442 


o636 


0829 


1023 


1216 


1410 


i6o3 


1796 1989 
3724! 3 9 i6 


i 9 3 


225 


2i83 


2375 


2568 


2761 


2954 


3i47 


3339 


3532 


i 9 3 


226 


4108 


43oi 


4493 


4685 


4876 


5o68 


5260 5452 


5643! 5834 


192 


:2 7 


6026 


6217 


6408 


6099 


6790 


6981 


7172 7363 7554^ 7744 


191 


223 


79 35 


8i25 


83i6 


85o6 


8696 


8886 


9076 92661 9456 ! 9646 


X 


229 


9'J35 


®®25 


•2l5 


•404 


©593 

2482 


•733 


•972,' 1 161 


i35o 


1039 


23o 


361728 


1917 


2105 


2294 


2671 


2859I 3o48 


3236 


3424 


188 


23l 


36i2 


3800 


3 9 S8 


4176 


4363 


455 1 


4739J 4926 


5n3 


53oi 


188 


232 


5438 


5675 


5862 


6049 


6236 


6423 


6610 6796 


6 9 83 


7169 


187 


233 


7356 


7542 


7729 


790 


8101 


8287 


8473 


865o 


8845 


9o3o 


186 


234 


9216 


9401 


9087 


9772 


9 9 58 


®i43 


•328 


•5i3 


®6 9 3 


•883 


i85 


235 


371068! 1253 


1437 


1622 


1806 


1991 


2175 


236o 


2544 


2728 


184 


236 


2912 


3096 


328o 


3464 


3647 


3b3i 


401 5 


4198 


4332 


4565 


184 


23 7 


4743 


4932 


5u5| 5298 


5481 


5664 


5846 


6029 


6212 


6894 


1 83 


238 


6J77 


6759 


6942 7124 
8761 1 8943 


7306 


7488 


7670 


7852 


8o34 


8216 


182 


239 


83 9 8 


85So 


9124 


93o6 


9487 


9668 


9849 


•®3o 


181 


240 


38021 1 


0392 


0573J 0754 


0934 


ni5 


1296 


U76 


1 656 


i83 7 


181 


241 


2017 


2197 


2377 2 55 7 


2737 


2917 


3o 97 


3277 


3456 


3636 


180 


242 


33 1 5 


3995 


4i74 


4353 


4533 


4712 


4891 


5070 


5249 


5428 


$ 


243 


56o6 


5 7 85 


5964 


6142 


632i 


6499 


6677 


6856 


7034 


7212 

89S9 


244 


7390 


7568 


7746 


7923 


8101 


8279 


8456 


8634 


8811 


173 


245 


9166 


9 343 


9020 


9698 


9875 


®®5i 


•228 


•4o5 


®582 


• 7 5 9 


$ 


246 


390935 


11 12 


1283 


1464 


1641 


1817 

35 7 5 


1993 


2169 


2345 


2521 


247 


2697 


2873 


3o48 


3224 


3400 


3pi 


3926 


4101 


4277 


176 


248 


44 J 2 


4627 


4802 


4977 


5i52 


5-326 


55oi 


56 7 6 


585o 


6025 


i 7 5 


249 


6199 


63 7 4 


6543 


6722 


6^96 


7071 
8808 


7245 


7419 


7 5 9 2 


7766 


174 


25o 


397940 


8114 


8287 


8461 


8634 


8981 


9154 


9 3 2 3 


95oi 


17* 


25l 


9674 


9847 


ce 20 


•192 


•365 


•538 


•711 


•833 


io56 


1228 


i 7 3 


232 


401401 


1573 


1745 


IQI7 


20S9 


2261 


2433 


26o5 


2777 


2949 


172 


253 


3i 2 1 


3292 


3464 


3635 


38o 7 


3 97 S 


4U9 


4820 


4492 


4663 


171 


2 54 


4834 


5oo5 


6176 


5346 


55i7 


5688 


5853 


6029 


6199 


6370 


Hi 


255 


654o 


6710 


68S1 


7o5i 


7221 


7891 


7 56i 


77-3i 


9 5 9 5 
1283 


8070 


170 


256 


8240 


8410 


85 79 


8749 


8918 


9087 


9207 


9426 


9764 


169 


23 7 


9933 


®I02 


•271 


©44o 


®6oo 
2293 


®777 


•946 


IH4 


U5i 


,69 


258 


411620 


I783 


1956 


2r24 


2461 


2629 


2796 


2964 


3i32 


168 


2 5g 


33oo 


3467 


3635 


38o3 


3970 


4i37 


43o5 


4472 


4639 


4806 


167 


260 


414973 


5i4o 


53o7 


5474 


564i 


58o8 


5974 


6141 


63o8 


6474 


167 


261 


6641 


6807 


6973 


7i3g 


7806 


7472 


7638 


.7804 


7970 


8i35 


166 


262 


83oi 


8467 


8633 


8798 


8964 


9129 


9 2 9 5 


9460 


9625 


979 1 


i65 


263 


9 rp6 


®I2I 


©286 


®45i 


®6i6 


®7;si 


®Q45 


1 1 10 


1275 


1439 


i65 


264 


421604 


I7 S8 


i 9 33 


2097 


2261 


2426 


2^90 


2754 


2918 


3o82 


164 


265 


3246 


34io 


35 7 4 


3/3 7 


3ooi" 


- 406 5 


4228 


4392! 4555 


4718 


164 


266 


4S82 


5o45 


5208 


53 7 i 


5d34 


5697 


586o 


6o23 ; 6186 


6349 1 • 63 


267 


65 1 1 


6674 


6836 


6990 


7161 


7324 


7486 


7648 7811 


7073 


162 


268 


8i35 


8297 


845o- 


8621 


8 7 83 


8944 


9 1 06 


9268 9429 


9 5 9 i 


162 


269 


9752 


9914 


©e 7 D 


®236 


•398 


•559 


•720 


•881! 1042 


i?o3 


.161 


270 


43 1 364 


i525 


i685 


1846 


2007 


2167 


2328 


2488; 2649 


2809 


161 


271 


2969 


3i3o 


3290 


345o 


36io 


3770 


3930 


4090 4249 


4409 


.160 


272 


4569I 47^9 


4888 


5o48 


5207 


5367 


5526 


5685, 5844' 6004 


1 59 


2 7 3 


6l63; 6322 


6481 


6640 


6798 


6 9 5y 


71 16 


7275 7433! 7592 


no 


274 


7 7 5i; 7909 


8067 8226 


8384 


8542 


8701 


88 59 901 7 9175 


1 58 


1 7 5 


9333 i 9491 


9648 


9806 


9964 


•122 


•279 


•437 »5 9 4i «7^ 


i58 


276 


440909; 1066 


1224 


i38i 


i538 1695 


1852 


2009 2166, 2323 


07 I 


277 


2480! 2637 


2793 


2950 


3io6l 3263 


3419 


J376| 3 7 32; 388 9 


• 5 7 | 


i 7 8 


4045 4201 


4357 


45i3 


4669 4825 


4981 


5i37| 5293- 5449 


i56 1 


a 79 


56o4 


5760 


5 9 j5 


6071 


6226 

4 1 


6382 


6537 


60921 6848 


7oo3 


i55 


1 





1 


__lJ 


: jl 


5 1 


6 


7 i 8 




I'D 





A TABLE 


OF 


LOGARITHMS FROM 


1 ro 


10,000. 


5 


N. 
280 


I 1 2 


3 1 4 | 5 j 6 7 1 8 


9 


D. 


447 1 58 ' 73i3' 7468 


7623. 7778J 7933 8088 8242 83^7 


8552 


i55 


281 


8706 1 8861 90 1 5 


9170 9324 9478 9633J 9787 


9941 


o© 9 5 


1 54 


282 


45o249i o4c3; o557 


07 1 1 1 o865| 1018 1 172! i326 


1479 


i633 


i54 


283 


1786' 1 9401 2093 


2247I 2400J 2553' 2706) 2859 


3oi2; 3i65 


1 53 


284 


33i8 347 1 1 3624 


3777! 3g3oi 4082 4235: 4387 


4340 


4692 


1 53 


285 


4845 


4997 ! 5i5o 


53o2; 5454 56o6 5758] 6910 


6062 


6214 


132 


286 


6366 


65i8: 6670 


6821 6973' 7125; 7276! 7428 


7 5 79 


773i 


l52 


2 ^ 


7882 


8o33 8184 


8336! 8487 j 8638| 8789' 8940 


9091 


9242 


i5i 


288 


9 3 9 2 


g543 9694 


9845; 999D1 e i46 °296| ©447 


®5 97 


a 748 


i5i 


289 


460898 


1048J 1 198 


i348 i499i ^49 '799 1948 


2098 


2248 


i5o 


ago 


462398 


2548j 2697 


2847 2997; 3 146 3296 3445 


35g4 
5o85 


3744 


i5o 


291 


33 9 3 


4042! 4191 


434o: 4490 4639 4788, 4 9 36 


5234 


149 


292 


5383 


553 2 1 568o 


582 9 ; 5 977 


6126, 6274; 6423 


65 7 i 


6719 


\it 


2g3 


6868 


7016 7164 


73 1 2 | 7460 


7608 7756 7904 


8o52 


8200 


294 


8347 


8495 8643 


8790 8g38 


9085 9233! 9 33o 


9 5 27 


9675 


148 


295 


9822 


9969 °u6 


®263l # 4io 


•557 «7o4 °85i 


* 99 8 


1 U5 


147 


296 


471292 


1438 i585 


1732! 1878 


2025 2171 23l8 


2464 


2610 


146 


297 


2 7 56 


2903 i 3 049 
4362J 45o8 


3i95| 334i 


3487! 3633 3770 


3925 


4071 


146 


298 


4216 


4653 4799 
6107 62D2 


4944^ 5090! 5235 
6397: 6542! 6687 


538i 


5526 


146 


299 


5671 


58i6 5g62, 


6832 


6976 


145 


3 00 


477121 


7266, 7411 


7555, 7700 


7844! 7989! 8i33 


8278 


8422 


u5 


3oi 


8566 


8711 8855^ 


8999! 9143 


9287: 9431 9575 


9719 


9 863 


144 


3o2 


480007 


oi5ii 0294' 


o438 o582 


072^ 08691 1012 


n56 


1299 


144 


3o3 


1443 


i586j 1729! 


1872 2016 


2l5o 2302 2445 


2588 


2 7 3l 


143 


3o4 


2874 


3oi6 3 1 5o 


33o2 3445 


358 7 ! 3 7 3o 38 7 2 


4oi5 


4157 


143 


3o5 


43oo 


4442| 453oi 


4727 4869 


5ou 5i53j 5295 


543 7 


5579 


142 


3o6 


5721 


58631 6oo5j 


6147 


6289 


643o, 6572! 6714 


6855 


6 9<n r 142 


307 


7i38 


7280 7421 


7563 
8974 


7704 


7845; 79S6 1 8127 


8269 


8410 


ui 


3o8 


855i 


8692 8333| 


91U 


9255' 9396; 9537 


9677 


9818 


141 


309 


99 58 


••99 


• 2 3 9 ; 


•38o e 52o 


•661 °8oi 


•941 


1081 


1222 


140 


3io 


491362 


l502 


16421 


1782 1922 


2062 ; 2201 


2341 


2481 


2621 


140 


3u 


2760 


2900 


3o4o 


3 1 7 9 ! 33i9 


34-53, 35 9 7 
485o ! 4989 


5 1 23 


38 7 6 


401 5 


i3 9 


3l2 


4i55 


4294 


4433 


4572; 4711 


5267 


5406 


i3 9 


3i3 


5544 


5633 


5822! 


5960' 6099 


6233 


63 7 6 


65 1 5 


6653 


6791 


i3q 


3 14 


6930 


7068 7206! 
8448 8536! 


7344 7483 


7621 


77 5 9 


7897 


8o35 


8i 7 3 


i38 


3i5 


83n 


8724 1 8862 


8999 


9 i3 7 


9275 


9412 


955o 


i38 


3i6 


9687 


9824; 9962! 


•099 «236 


•374 


•5n 


•648 


®735 


•922 


i3 7 


317 


5oio59 


1196 i333| 


1470J 1607 


1744 


iSSo 


2017 


2 1 54 


2291 

3655 


i3 7 


3i8 


2427 


2564 


2700: 


2837I 2973 


3109 


3246 


3382 


35i8 


i36 


319 


3791 


3 9 2 7 


4o63 


4199 4-335 


4471 


4607 


4743 


4873 


5oi4 


i36 


320 


5o5i5o 


5286 


542 1 1 


5557 5693 


5828 5 9 64 


6099 


6234 


6370 


i36 


321 


65o5 


6640 


67761 


6911 7046 


7181 


7 3i6 


745i 


7586 


7721 


i35 


322 


7856 


7991 


81261 


8260 8395 


853o 


8664 


8799 


8 9 34 


9068 


i35 


323 


92o3 


9337 


9471 


9606 9740 


9874 


••e 9 


oi43 


•277 


041 1 


1 34 


324 


5io545 


0679 °3i3 


0947 1081 


I2l5 


1 349 


1482 


1616 


1750 


i34 


325 


i883 


2017; 2 1 5 1 J 


2284 2418 


255i 


2684 


2818 


2951 


3o84 


i33 


326 


3218 


335i 


3484J 


3617 375o 


3883 


4016 


4149 


4282 


4414 


i33 


32 7 


4548 


-463 1 


48i3 


49461 5079 


52 r 1 


5344 


5476 


56o 9 


5741 


i33 


328 


58 74 


' 6006 


6i3qi 


6271 64o3 


6535 


6668 


6800 


6932 


7064 


132 


3^ 


c V? 6 


7328; 7460J 


7D92J 7724 


7355 


79 37 


8119 


825i 


8382 


i3a 


33o 


5i85i4 


8646 87771 


89091 9040 


9171 


9,3o3 


9434 


9 566 


9697 


i3i 


33 1 


9828! 99 5 9 ; •• 90| 


°22i| e 353 


e 484 


«6i5 


•743 


•876 


1007! i3 1 


332 


5jii3S, 1269 1400 


i53o 1 66 1 


1792 


IQ22 


2 353 


2i83 


23i4 i3i 


333 


2444! 2575 2705 


2835 t 2966 


3o 9 6 3226 


3356 


3486 


36i6 i3o 


33; 


3746' 3876; 4006 


4 1 36 ; 4266 


4396I 4526 


4656 


4 7 85 


491 5| i3o 


335 


5o45| 5i74l 53o4 


5434.' 5563 


56 9 3 582 2 


5 9 5i 


60S 1 


6210 129 


336 


633g 6469 65q3 


6727 6856 


6985 7114 


7243, 


7372 


75or 129 


33 7 


763o: 7759 78881 


8016 8i45 


8274J 8402 


853 1 


8660 


8788 I2Q 


338 


8917J 9045 9174 


9302 i 943o 


9539; 9687 


9S15; 9943 


••72' 128 


33 9 


53 02 00 


o328 


0456 


o584 


0712 


0840 


008 


1096 


1223 


1 35 1 


128 



UL 



l > I 



! 9 i 



KJ 



G 


A TABLE 


OF LOGA 


UTHMS FROM X 


to 10,000. 




N. 





I 


2 


3 


4 


5 


6 1 7 j 8 


9 


D. 


340 


531479 


1607 


1734 


1862 


1990 


2117 


2245 2372! 25oo 


2627 


I28" 


34i 


2754 


2882 


3009 


3i36 


3264 


3391 


35i8i 3645' 3772 


38 99 


127 


J42 


4026 


41 53 


4280 


4407 


4534 


4661 4787: 4914 5o4i 


5167 


127 


343 


D294 


5421 


5547 


5674 


58oo 


5927 6o53 6180 63o6 


6432 


126 


344 


6558 


6685 


681 1 


6937 


7063 


7189' 73i5| 7441 7^67 


7693 


126 


345 


7819 


7945 


8071 


8197 


8322 

9 5 7 8 


8448 85 7 4i 8699 8825 


8951 


126 


346 


.9076 


9202 


9 32 7 


9452 


97 o3 


9829! 9984 °°79 


•204 


125 


347 


540329 


0455 


o58o 


0705 


o83o 


0955 


1080 i2o5 i33o 


1454 


125 


348 


1 D79 


1704 


1829 


1953 


2078 


2203 


2327I 2452 2576 


2701 


125 


349 


2825 


2950 


3074 


3i 99 


3323 


3447 


3571 


36 9 6, 382c 


3944 


124 


3jo 


544o68 


4192 


43i6 


444o 


4564 


4688 


4812 


4936 5o6o 


5i83 


124 


35i 


5307 


543 1 


5555 


5678 


5So2 


5 9 25 


6049 


6172; 6296 


6419 


124 


352 


6543 


6666 


6789 


6913 


7o36 


7i5 9 


7282 


74o5 7029 
8635 8 7 58 


7652 


123 


3j3 


777 5 


7898 


8021 


8144 


8267 


838 9 


85i2 


8881 


123 


334 


9003 


9126 


9249 


9871 


9494 


9616 


97 3 9 


.9S61 9984 


•106 


123 


3-35 


5502 28 


o35i 


0473 


OJg5 


0717 


0S40 


0962 


1084 1206 


i328 


122 


356 


i45o 


1572 


1694 


1816 


1938 


2060 


2181 


23o3! 2428 


2547 


122 


35 7 


2668 


2790 


291 1 


3o33 


3i55 


3276 


33 9 8 


3819, 364o 


3762 


121 


358 


3883 


4004 


4126 


4247 


4368 


4489 


4610 


473i! 4852 


4973 


121 


359 


5094 


52 I 


5336 


5457 


55-78 


56 99 
6 9 o5 


5820 


59401 6061 


6182 


121 


36o 


5*>3o3 


6423 


6544 


6664 


6785 


7026 


7146! 7267 


738 7 


120 


36i 


7 5o 7 


7627 


7748 


7868 


7988 


8108 


8228 


8349| 8469 


8589 


120 


36a 


8709 


8829 


894S 


9068 


9188 


93o8 


9428 


95481 9667 


9787 


120 


363 


9907 


•026 


•146 


•265 


•385 


c 5o4 


•624 


743! ©863 


•982 


119 


364 


56 1 101 


1221 


1 34o 


U59 


1078 


1698 


1817 


1936, 2o55 


2174 


II9 


365 


2293 


2412 


253 1 


265o 


2769 


2887 


3oo6 


3i25i 3244 


3362 


119 


360 


» 348i 


36oo 


3 7 i8 


3837 


3955 


4074 


4192 


43 nl 4429 


4548 


119 


36 7 


4666 


4784 


4903 


5021 


5 1 39 


5257 


5376 


5494, 56 1 2 


5 7 3o 


Il8 


368 


5848 


5966 


6084 


6202 


6320 


6437 


6 358 


6673; 679; 


6909 


:i8 


36 9 


7026 


7144 


7262 


7379 


7497 


76i4 


7782 


78491 7967 


8084 


uS 


3 7 o 


568202 


83 1 9 


8436 


85J4 


8671 


8788 


890J 


9023: 01 4o 


9257 


117 


3 7 i 


9 3 74 


9491 


9608 


9725 


9842 


99:39 


«o 7 6 


•i93| °3og 


°426 


117 


3 7 2 


570543 


0660 


0776 


0893 


1010 


1 1 26 


1243 


i359 1476 


1892 


117 


3 7 3 


1709 


1825 


1942 


2o58 


2174 


2291 


2407 


2523! 263g 


2755 


116 


374 


2872 


2988 


3 1 04 


3220 


3336 


34J2 


3568 


3684 i 38oo 


3 9 ,5 


116 


3 7 5 


4o3i 


4i47 


4263 


4379 


4494 


4610 


4726 


4841 1 4957 


5072 


116 


3 7 6 


5 1 88 


53o3 


5419 


5534 


565o 


5 7 65 


568o 


5996: 61 1 1 


6226 


n5 


3 77 


634i 


6457 


6J72 


6687 


6802 


6917 


7032 


7147 7262 


7377 


1 15 


3 7 .8 


7492 


7607 


7722 


7836 


7 9 5 1 


8066 


8l8l 


8295 8410 


8525 


n5 


379 


863g 


8 7 54 


8868 


8983 


9097 


9212 


9326 


9441 ] 9555 


9669 


114 


3oo 


579784 


9898 


•0,2 


•126 


•241 


•3:35 


*46 9 


•583 ©697 


•Sn 


114 


38i 


580925 


1039 


1 1 53 


1267 


i38i 


1493 


1608 


1722, i836 


1950 


114 


332 


2o63 


2177 


2291 


2404 


25i8 


263 1 


2745 


28188 2972 


3o85 


114 


383 


3199 


33i2 


3426 


3539 


3652 


3765 


3879 


3992 4io5 


4218 


n3 


334 


433 1 


4444 


4557 


4670 


4783 


4896 


5009 


5i22 5235 


5348 


n3 


335 


5461 


5574 


5686 


5799 


5912 


6024 


6i3 7 


625o 6362 


6475 


n3 


3S6 


6587 


6700 


63i2 


6923 


7 o3 7 


7 '49 


7262 


7374 7486 


7299 


112 


337 


7711 


7823 


79 35 


8047 


8160 


8272 


8384 


8496 8608 


8720 


1 12 


I 3oS 


8832 


8944 


9o56 


9167 


9279 


9 3 9 i 


9803 


96i5 9726 


9 838 


112 


3! 9 


9960 


•°6 1 


°i73 


•2S4 


•3 9 6 


•507 


•619 


°73o ©842 


• 9 53 


112 


390 


59 1 o65 


1 176 


1287 


1 399 


i5io 


162 1 


I 7 32 


1843 1955 


2066 


1 11 


391 


2177 


2288 


2399 
35o8 


2)10 


2621 


2732 


2843 


2954 3o64 


3 175 


in 


3g2 


3286 


33 97 


36i8 


3729 


3840! 3 9 5o 


4061 4171 


4282 


111 i 


3 9 3 


43 9 3 


45o3 


4614 


4724 


4834 


4945 5o55 


5i65 • 5276 


5386 


no i 


3 9 4 


5496 


56o6 


5 7 i 7 


5827 


5 9 3 7 


6047 1 6,5 7 


6267 6377 


6487 


.10 | 


395 


65 97 


6707 


6817 


6927 


7037 


7146 7256 


7366 7476 


7586 


110 | 


396 


7t 9 5 


78o5 


79i4 


8024 


8i34 


8243' 8353 


8462 85 7 2 


8681 


no | 


397 


879. 


8900 


9009 


9119 


9228 


9337 9446 


9886 9665 
•646 °755 


9774 


109 


398 


9883 


999 2 


°I0I 


•210 


•319 


•428 ©537 


•864 


109 


399 

"n. 

1 1 


600973 


10821 nqi 


12Q9 


1408 


i5n 1620 


1734 io43i ig5i 


109 





. ' !■ » 


3 ! 4 


5 ; 6 


~7~!~T" 


9 


D. 



A. TABLE OF LOGARITHMS FROM 1 TO 1U.000. 



N. 

400 


__°1 


■ 


2 

2277 


3 

2386 


4 


5 


6 


' 


8 


9 


D. 


602060! 2169 


2494, 26o3 


2711 


2819 


2928 


"3o36' J 08 


401 


3i44 3253 


336i 


3469 


3577 


3686 : 3794 


3902 


4010 41 18 j 08 


402 


4226| 4334 


4442 


455o 


4658 


4766! 4874 


4982 


5089 


5l 97 : I08 


4o3 


53o5 


54i3 


552i 


5628 


D736 


5844 


0951 


6o5g 


6166 


6274 I08 


404 


638i 


6489 


6596 


6704 


681 1 


6919 


7026 


7i33 


7241 


7348: I07 
8419 1 107 


400 


7455 


7562 


7669 


7777 
8347 


7834 


799' 


8098 


8 2 o5 


83 1 2 


406 


8526 


8633 


8740 


8 9 54 


9061 


9167 


9 2 ~4 


9 38i 


94381 107 


407 


Q594 


9701 


9808 


9914 


e© 21 


•128 


•234 


•341 


«447 


'•554! 107 


408 


610660 


0767 


0873 


0979 


1086 


1192 


1298 


Mo5 


i5n 


1617, 106 


409 


1723 


1829 


i 9 36 


2042 


2143 


2254 


236o 


2466 


2572 


2678J 106 


410 


612784 


2890 


2996 


3 1 02 


3207 


33i3 


3419 


3525 


363o 


3-36 1 106 


4U 


3842 


3 9 47 


4od3 


4i5g 


4264 


4370 


4473 


458i 


4686 


479 2 


io6 


412 


4897 


5oo3 


5io8 


5 2 i3 


53i 9 


5424 


5529 


5634 


5740 


58 4 5 


io5 


4i3 


' 59D0 


6o55 


6160 


6265 


6370 


6476 


658i 


6686 


6790 


68 9 5 


io5 


4U 


7000 


7io5 


7210 

8 2 5 7 


73 1 5 


742o 


llf, 


7629 
8676 


7734 


7 83 9 


7943 


io5 


41 5 


8048 


8i53 


8362 


8466 


8780 


8S84 


8989 


io5 


416 


90g3 


9198 


9302 


9406 


95i 1 


961 5 


9719 


9824 


9928 


003 2 


104 


417 


620136 


0240 


o344 


0448 


o552 


o656 


0760 


0864 


0968 


IO72 


104 


418 


1 1 76 


1280 


1 384 


1488 


1592 


i6o5 
2732 


1799 


1903 


2007 


21 10 


104 


419 


2214 


23i8 


2421 


2525 


2628 


2835 


29J9 


3o42 


3l46 


104 


420 


623249 


3353 


3456 


3559 


3663 


3766 


3869' 


3973 


4076 


4179 


io3 


421 


4282 


4385 


4488 


4591 


46 9 5 


4798 


4901 


5oo/, 


5(07 


5210 


io3 


422 


53i2 


54i5 


55i8 


562i 


5724 


5827 


5929 


6o32 


6i35 


6238 


io3 


423 


6340 


6443 


6546 


6648 


6751 


6853 


6956 


7o58 
8082 


7161 
8i85 


7263 


io3 


424 


7366 
838 9 


7468 
8491 


7571 
85 9 3 


7673 
8695 


7775 


7878 


7980 


8287 


102 


425 


8797 
9817 


8900 


9002 


9 1 04 


9206 


93o8 


102 


426 


9410 


o5i 2 


9613 


97 1 5 


9919 


°®2I 


©123 


°224 


«326 


102 


427 


630428J o53o 


o63i 


0733 


o835 


0936 


io38 


1 139 


I 24l 


1 342 


102 


428 


1444 


1045 


1647 


1743 


1849 


1951 


20D2 


2 1 53 


2255 


2356 


101 


429 


245 7 


2559 


2660 


2761 


2862 


2963 


3o64 


3i65 


3266 


336 7 


101 


43o 


633468 


3569 


36 7 o 


3771 


3872 


3973 


4074 


4n 5 


4276 


43 7 6 


100 


43 1 


4477 


45 7 8 


4679 


4779 


4880 


4981 


5o8i 


5i32 


5283 


5383 


100 


432 


5434 


5584 


5685 


5 7 85 


5886 


5 9 86 


6087 


6187 


6287 


6388 


loo 


433 


6488 


6588 


6688 


6789 


6889 


69S9 


7089 
8090 


7,89 
8190 


7290 


73oo 
838 9 


ICO 


434 


7490 
8489 


j5ao 
8389 


7690 

&6»9 


779°' 
87% 


7890 

8888 


7990 


8290 


99 


435 


8 9 88 


9088 


9188 


9287 


938 7 


99 


436 


94S6 


9586 


9686 


9783 


9 885 


9984 


»®84 


•i83 


•283 


©382 


99 


43-7 
433 


640431 


o58i 


0680 


0779 


0879 


0978 


1077 


1177 


1276 


i3 7 5 


99 


1474 


1573 


1672 


1771 


1871 


1970 


2069 


2168 


2267 


2366 


99 


43 9 


2465 


2563 


2662 


2761 


2860 


2959 


3o58 


3i56 


3255 


3354 


2 


44o 


643453 


355i 


365o 


3749 


3847 


3946 


4044 


4143 


4242 


434o 


44 1 


4439 


4537 


4636 


4734 


4832 


493i 


5029 


5i 27 


52 26 


5324 


98 


442 


5422 


552i 


5619 


5717 


58 1 5 


5oi3 


601 1 


61 10 


6208 


63o6 


98 


443 


6404 


65o2 


6600 


6698 


6796 


6894 


6992 


7o3 9 
8067 


7187 


7285 


*? 


444 


7383 


7481 
8458 


7579 


7676 


7774 


7872 


7969 


8i65 


8262 


98 


445 


836o 


855d 


8653 


8 7 5o 


8848 


8943 


9043 


9140 


9237 


97 


446 


9 335 


943 a 


953o 


9627 


9724 


9821 


,99'9 

0890 


••16 


•u3 


•210 


97 


447 


65o3o8 0405 


0D02 


0599 


0696 


0793 


0987 


1084 


11S1 


97 


443 


1278 


i3 7 5 


U72 


1 569 


1666 


1762 


i85 9 


1956 


2o53 


2t5o 


97 


44 9 


2246 


2343 


2440 


2536 


2633 


273o 


2826 


2923 


3oi 9 


3i:6 


97 


45o 


6532i3 


3309 
4273 


34o5 


35o2 


35 9 8 


3695 


3791 


3888 


3984 


4080 


96 


45i 


4177 


436 9 


4465 


4562 


4658 


47^4 


485o 


4946 


5o42 


96 


4'3 2 


5i33 


5235 


533 1 


5427 


5523 


5619 


57i5 


58 1 6 


5906 
6864 


6002 96 


453 


6098 


6194 


6290 


6386 


6482 


6577 


6673 


6769 


6960, 96 


454 


7o56 


7i52 


7247 


7343 


7438 


7534 


7629 


7725 


7820 


79 ,6 i 9 6 


455 


801 1 


8107 


8202 


8298 


83 9 3 


8488 


85S4 


8679 


8774 


8870 


9 5 


456 


8 9 65 


9060 


qi55 


925o 


9346 


944i 


9 536 


9 63 1 


9726 


9821 


9 5 


437 


9916 


®»ii 


°io6 


°20I 


0296 
1245 


0391 


®486 


©5Si 


•676 


0771 


9 5 


458 


66o865 


0960 


io55 


I i5o 


i339 


1434 


1029 


1623 


1718 


95 


459 


i8i3 


1907 


2002 


2096' 219! 


2286' 238o 


2475 


2569 


2663 95 


N. 





' 


2 | 3 J 4 1 5 j 6 


7 


8 


9 i ix 



8 


A TABLE 


OF 


LOGARITHMS FROM \ 


TO 


10,000. 




N. 





1 2 | 3 | 4 | 5 | 6 | 7 J 8 1 9 | D. 


460 


662758 


2852 2947I 3o4i 3i35 323o; 3324 3418 35i2 ! 3607I 94 


461 


3701 


37901 388 9 ! 3983 : 4078' 4172 4266 436o, 4454| 4548| 94 


462 


4642 


4736| 483o 


4924 5oi8 


1 DI12J 5206; 5299! 5393 5487I 94 


463 


558 1 


5675 5769 


5862 5956 


6o5o 6143 6237 633i 6424 94 


464 


65i8 


6612 


6705 


6799 1 6892 


6986 7079; 7173 7266I 736o 


94 


465 


7453 


7546 


7640 


7733 7826 


7920 


8oi3 8io6| 8199! 8293 


9 3 


466 


8386 


8479 


8572 


8665 


8759 


8852 


8945! 9038 9131 9224 


9 3 


467 


< 93l l 


9410 


95o3 


9596 


9689 


9782 


9875 


9967 1 *»6o 


°i53 


9 3 


468 


670246 


o339 


0431 


o524 


0617 


0710 


0802 


0895I 0988 


1080 


93 


469 


1 173 


1265 


i358 


I45i 


1 543 


1636 


1728 


1 82 1 lgi3 


2005 


93 


470 


672098 


2190 


2283 


2375 


2467 


256o 


2652 


2744 2836 


2929 


92 


47i 


3021 


3n3 


32o5 


3297 


3390 


3482I 3574 


3666 3758 


385o 


92 


472 


3o42 

4061 


4o34 


4126 


4218 


43 10 


4402 


4494 


4586 4677 


4769 


92 


473 


4g53 


5o45 


5i3 7 


5228 


5320 


5412 


55o3[ 55g5l 5687 


92 


474 


5778 


5870 


5962 


oo53 


6145 


6236 


6328 


6419 


65n 6602 


92 


473 


6694 


6 7 85 


6876 


6968 


7o5 9 


7 1 5i 


7242 


7 333 


7424! 7 5l 6 


9i 


476 


7607 
85i8 


7698 


7789 


7881 


7972 


8o63 


8i54 8245 


8336; 8427 


9i 


477 


8609 


8700 


8791 


8882 


8 97 3 


9064; 9 1 55 9246 


9 33 7 


9 1 


478 


9428 


9 5l 9 


9610 


9700 


979 1 


9882 


997 3 «o63 •i54 


•245 


9i 


479 


68o336 


0426 


o5i7 


0607 


0698 


0780 0879! 0970 1060 


1 i5i 


9i 


480 


681241 


i332 


1422 


i5i3 


i6o3 


1693 1784! 1874 1964 


2o55 


90 


481 


2i45 


2235 


2326 


2416 


25o6 


2596 1 2686 


27771 2867 


2 9 5 7 


90 


482 


3o47 


3i3 7 


3227 


33i 7 


3407 


3497 


358 7 


36 7 7 3767 


385 7 


90 


4S3 


3947 


4o37 


4127 


4217 


43o7 


43 9 6 


4486 


4576! 4666 


4756 


90 


484 


4845 


4935 


5o?5 


5ii4 


52o4 


5294 


5383 


5473; 5563 


5652 


90 


485 


5742 


583 1 


5921 


6010 


6100 


6189 


6279 


6368| 6458 


6547 


89 


486 


6636 


6726 


68i5 


6904 


6994 


7oS3 


7172 


7261 735i 
8i53 ! 8242 


744o 


89 


487 


7529 
8420 


7618 


7707 


V? 6 


7806 


7975 


8064 


833 1 


2 9 


488 


8509 


85 9 S 


8687 


8776 


8865 


8o53 
9S41 


9042 9 i3i 
99 3o ••i 9 


9220 


^ 


489 


9309 


9 3 9 8 


9486 


9575 


9664 


97 53 


©107 


o 9 


490 


690196 


0285 


o373 


0462 


o55o 


o63g 


0728 


0816 0905 


^93 


11 


491 


1081 


1170 


1258 


l347 


1435 


i524 


161 2 


1700 1789 


1877 


492 


1965 


2o53 


2142 


2230 


23i8 


2406 


2494 


2583 2671 


2 7 5 9 


88 


4g3 


2847 


2g35 
38i5 


3o23 


3i 1 1 


3199 


3287 


3375 


3463 355i 


363 9 


88 


494 


3727 


3903 


3991 4078 


4 1>66 


4254 


4342! 443o 


45i7 


88 


4g5 


46o5 


4693 


4781 


4868 


4956 


5o44 


5i3i 


5219' 53o7 


5394 


88 


496 


0482 


556g 


5657 


5744 


5832 


5 9 i 9 


6007 


6094^ 6182 


6269 


2 7 


497 


6356 


6444 


653 1 


6618 


6706 


6 79 3 


6880 


6968! 7o55 


7U2 


87 


498 


7229 


7317 


74o4 


7491 


7578 
8449 


7665 


7 7 52 


783 9 


7926 


8014 


87 


499 


8101 


8188 


8275 


8362 


8535 


8622 


8709 


8796 


8883 


^ 


5oo 


69S970 


9 o5 7 


9U4 


9231 


9317 


9404 


9491 


9 5 7 8 


9664 


975i 


^ 7 


5oi 


9 838 


9924 


••11 


••98 


•184 


•271 


•358 


444 


•53 1 


•617 


87 


502 


700704 


0790 


0877 


0963 


io5o 


n36 


1222 


1 309 


1395 


1482 


86 


5o3 


1 568 


1604 


I74i 


1827 


Jgi3 


1999 
2861 


2086 


2172 


2258 


2344 


86 


5o4 


243 1 


25i7 


26o3 


2689 


2 77 5 


2947 


3o33 


3i 19 


32o5 


86 


5o5 


3291 


33 77 


3463 


3549 


3635 


3721 


3807 


38 9 3 


3979 


4o65 


86 


5o6 


401 


4236 


4322 


4408 44g4 


4579 


4665 


475i 


483 7 


4922 


86 


507 


5oo8 


5094 


5179 


5265 


535o 


5436 


5522 


5607 


56 9 3 


5 77 8 


86 


5oS 


5S64 


5g49 
68o3 


6o35 


6120 


6206 


6291 


63 7 6 


6462 


6547 


6632 


85 


5og 


6718 


6SSS 


6974 


7059 


7144 


7229 


7 3i5 


74oo 


7485 


85 


5io 


707J70 


7655 


7740 


78261 791 1 


7996 


8081 


8166 


825i 


8336 


85 


5u 


8421 


85o6 


8591 


8676 8761 
9524 9609 


8846 893 1 1 9015 


9100 


9185 


85 


DI2 


9270 


9355 


9440 


9694 9779 9 863 


9948 


°«33 


85 


5i3 


710117 


0202 


0287 


0371 0456 


o540j 0625 


0710 0794 


0879 


85 


5i4 


0963 


1048 


Il32 


1217 i3oi 


1 385 1470 


i554 1639 


1723 


84 


5i5 


1807 


1802 


1976 


2060 2144 


22291 23i3 


2397. 2481 
3238 3323 


2566 


84 


5i6 


265o : 2734 


2818: 2902 2986 
365 9 3742! 3826 


3o 7 o 3 1 54 


3407 


84 


5l l 
5i8 


3491! 3575 


3910 3994 


4078 4162 


4246 


84 


433o: 4414 


4497 458ij 4665| 4749| 4833 


4916' Sooo, 5o84 


84 


519 


516-71 525i 


5335 54181 55o2 : 5586 566 9 


5753 5836 5920 




K. 


| , 


2 | 3 | 4 | 5 | 6 


7 i 8 I 9 J 





A TABLE 


OF 


LOGARITHMS Fl 


tOM 


I TO 


10,000. 


9 


N. 


1 I 1 2 


3 j 4 | 5 


| 6 ; 7 


8 ! 9 


D. 


5ao 


;i6oo3 6o8 7 [ 6170 


6254 6337 6421 


1 65o4 


1 6588 


667 1 1 6754 


s 


521 


6838 


692 1 1 7004 


,7088 7171 7254 


7338 


742i 


75o4i 7 5 87 
8336| 8419 


83 


522 


7671 


7734! 7837 


792c! 8oo3 ; 8086 


8169 


8 2 53 


83 


533 


85o2 


85851 8668 


873i| 8834' 8917 


9000 


9 o83 


916D, 9248 


83 


524 


9 33 1 


94141 9497 


95801 9663! 9745 


9828 


991 1 


9994 


•°77 


83 


525 


720159 


0242 


0323 


04071 0490 0573 


o655 


o 7 38 


0821 


0903 


83 


526 


0986 


1068 


ii5i 


1233 i3i6 1398 


1481 


1 563 


1646 


1728 


82 


52 7 


181 1 


1893 


i 97 5 


2o58 2140! 2222 


23o5 


2387 


2469 


'2552 


82 


52b 


2634 


2716 


2798 


288 1 i 2963 


3o45 


3127 


3209 


3291 


3374 


82 


! 52 9 


3456 


3538 


3620 


3702I 3784 


3866 


3948 


4o3o 


4112 4i94 


82 


53o 


724276 


4358 


4440 


4522! 4604 


4685 


4767 


4849 493 1 j 5oi3 


82 


53 1 


5095 


5176 


5 2 58 


5340 5422 


55o3 


5585 


56671 5748', 583o 


82 


532 


5912 


5993 


6075 


"61 56 


6238 632o 


6401 


6483; 6564i 6646 


82 


533 


6727 


6809 


6890 


6972 


7o53 7134 


7216 


7297 


7 3 79 


7460 


81 


534 


7541 


7623 
8435 


7704 


77 85 


7866J 7948 


8029 


8110 


8191 


8273 


81 


535 


8354 


85i6 


85 97 


8678 


8 7 5 9 


8841 


8922 


9003 


9084 


81 


536 


9165 


9246 


9327 


9408 


9489 


9370 


965i 


97 32 


9 8i3 


9893 


81 


53 7 


9974 


••55 


•i36 


•217 


©298 


•3 7 8 


•459 


•340 


•621 


•702 


81 


538 


730782 


o863 


0944 


1024 


no5 


1 186 


1266 


1 347 


1428 


i5o8 


81 


53 9 


1589 


1669 


1750 


i83o 


1911 


1991 


2072 


2l52 


2233 


23i3 


81 


540 


732394 


2474I 2555 


2635 


2715 


2796 


2876 


2956 


3o37 


3 1 17 


80 


541 


3197 


32 7 8| 3358 


3438 


35i8 


3598 


3679 


3759 


383 9 


3919 


80 


54^ 


3999 


4079 


4160 


4240 


432o 


4400 


4480 


4560 


4640 


4720 


80 


543 


4800 


4880 


4960 


5o4o 


5l20 


5200 


5279 


5359 


543 9 


55i9 


80 


544 


5599 


5679 


5 7 59 


5838 


5918 


5998 


6078 


6i57 


6237 


63i7 


80 


545 


6397 


6476 


6556 


6635 


6715 


6795 


6874 


6954 


7o34 


71 13 


80 


546 


7 i 9 3 


7272 


7352 


743i 
8225 


731 1 
83o5 


7590 
8384 


7670 


7749 


7829 


7908 


79 


54? 


7087 
8781 


8067 


8146 


8463 


8543 


8622 


8701 


79 


548 


8860 


8 9 39 


9018 


9097 


9177 


9256 


9 335 


9414 


9493 


79 


549 


9572 


9651 


97 3i 


9810 


9889 


9968 


•®47 


•126 


«205 


•284 


79 


55o 


74o363 


0442 


o52i 


0600 


0678 


0757 


o836 


0915 


0994 


1073 


79 


55i 


Il52 


1230 


i3o9 


i388 


1467 


1546 


1624 


1703 


1782 


i860 


79 


552 


i 9 3 9 


2018 


2096 

2882 


2175 


2254 


2332 


241 1 


2489 


2568 


2647 


8 


553 


2725 


2804 


2961 


3o39 


3n8 


3196 


32 7 5 


3353 


343 1 


554 


35io 


3588 


3667 


3745 


38 2 3 


3902 


3 9 8o 


4o58 


4i36 


42i5 


78 


555 


4293 


43 7 i 


4449 


4528 


4606 


4684 


4762 


4840 


4919 


4997 


78 


556 


5075 


5i53 


523i 


53o9 


5387 


5465 


5543 


562i 


5699 


5777 


78 


557 


5855 


5 9 33 


601 1 


6089 


6167 


6245 


6323 


6401 


6479 


6556 


78 


558 


6634 


6712 


6 79 o 


6868 


6945 


7023 


7101 


7 X 79 


7256 


7334 


78 


55 9 


7412 


7489 
8266 


7567 


7645 


7722 


7800 


7878 


79 55 


8o33 


8110 


78 


56o 


748188 


8343 


8421 


8498 


8376 


8653 


8731 


8808 


8885 


77 


56 1 


8 9 63 


9040 


9118 


9 i 9 5 


9272 


935o 


9427 


95o4 


9582 


9659 


77 


562 


9736 


9814 


9891 


9968 


•®43 


•123 


•200 


©277 


•354 


•43 1 


77 


563 


75o5o8 


o586 


o663 


0740 


0817 


0894 


0971 


1048 


1125 


1202 


77 


564 


1279 


i356 


1433 


i5io 


i587 


1664 


I74i 


1818 


i8 9 5 


1972 


77 


565 


2048 


2125 


2202 


2279 


2356 


2433 


2509 


2586 


2663 


2740 


77 


566 


2816 


2893 


297O 


3o47 


3i 2 3 


3200 


3277 


3353 


343o 


35o6 


77 


56 7 


3583 


366o 


3 7 36 


38i3 


3889 


3966 


4042 


4i 19 


4195 


4272 


77 


568 


4348 


4425 


45oi 


45 7 8 


4654 


4730 


4807 


4883 


4960 


5o36 


76 


569 


5ll2 


5i8 9 


5265 


534i 


5417 


5494 


5570 


5646 


5722 


5799 


76 


5 7 o 


755875 


5g5i 


6027 

6788 


6io3 


6180 


6256 


6332 


6408 


6484 


656o 


76 


071 


6636 


6712 


£864 


6940 


7016 


7092 


7168 


7244 


7320 


76 


572 


7396 


7472 


7548 


7624 


7700 


7775 


785i 


7927 


8oo3 


8079 


-6 


5 7 3 . 


8i55 


823o 


83o6 


8382 


8458 


8533 


8609 


8685 


8761 


8836 


-6 


5 7 4 


8912 


8988 


9o63 


9 i3 9 


9214 


9290 


9 366 


944i 


9517 


9592 


76 


073 


9668 


9743 


9819 


9894 


9970 


••45 


•121 


•196 


•272 


•347 


75 


5 7 6 


760422 


0498 0D73 


0649 


0724 


0799 


0875 


0950 


1025 


KOI 


75 


5 77 


1176 


I25l 


1326 


1402 


1477 


i552 


1627 


1702 


1778! 


1 853 


75 


5 7 8 


1928 


2oo3 


2078 


2i53 


22:8 23o3 


23 7 8 


2453 


2529 

3278J 


2604 


75 


! S79 


2679 


2 7 54 


2829 


2904 


2978 3o53 


3i28 


32o3 


3353 


73 


1 N. 





1 


2 


3 


4 


5 1 6 


7 


8 | 


9 


D. 



10 


A TABLE 


OF 


LOGARITHMS FROM 1 TO 


10,000. 




N. 


| , 


2 


3 | 4 


5 


6 


_L_ 


__L 


9 


D. 


58o 


763428 35o3 


35 7 8 


3653! 3727 


38o2 


~38ri 


3952 


4027 


4101 


75 


58 1 


4176! 425i 


4326 44oo 4475 


455o 


4624 


4699 


4774 


4848 


7 5 


582 


4923 499 8 


5072I 51471 522i 


6296 5370 


5445 


5520 


55 9 4 


75 


583 


566 9 5743 


58i8 


58 9 2; 5966 


6041 


6l ID 


6190 
6933 


6264 


6338 


74 


584 


64i3 


6487 


6562 


6636 6710 


6 7 85 


685 9 


7007 


7082 


74 


585 


71*56 


723o 


73o4 


7379i 7453 


7327 
8268 


760I 


7675 


7749 


7823 


74 


' 586 


7 8 9 8 


7972 


8046 


8120 8194 


8342 


8416 


8490' 8564 


74 


58-1 


8638 


8712 


8786 


88601 8 9 34 


9008 


9082 


9i56 


9 23o 9 3o3 


74 


588 


9377 


945i 


9323 


9399 9673 


9746 


9820 


9 8 9 4 


99681 «»42 


74 


58 9 


770115 


0189 


0263 


o336 


0410 


0484 


0537 


o63i| 0705 


0778 


74 


Sgo 


770822 


0926 


°999 


1073 


1146 


1220 


1293 


1 367I 1440 


i5i4 


74 


5gi 


1387 


1661 


1734 


1808 


1881 


1955 


2028 


2102J 2175 


2248 


73 


5g2 


2322 


23 9 5 


2468 


2542 


26i5 


2688 


2762 


2835 2 9 o8 


2981 


73 


5g3 


3o55 


3i28 


3201 


3274 


3348 


3421 


3494 


3567) 3640 


3 7 i3 


73 


5 9 4 


3786 


386o 


3 9 33 


4006 


4079 


4i52 


4225 


4298 4371 


4444 


73 


5 9 5 


45i7 


4390 


4663 


4736 


4809 


4882 


4933 


5o28 


5 1 00 


5n3 


73 


5 9 6 


5246 


5319 


53 9 2 


5465 


5538 


56io 


5683 


5 7 56 


58 29 


5902 


73 


5 97 


5 97 4 


6047 


6120 


6i 9 3 


6265 


6338 


641 1 


6483 


6556 


6629 


73 


5q8 


6701 


6774 


6846 


6919 


6992 


7064 


7137 


720 9 


7282 


7354 


73 


5 99 


7427 
778151 


7499 


7572 


7644 


7717 


7789 
85i3 


7862 


7934 


8006 


8079 


72 


600 


8224 


82 9 6 


8368 


8441 


8585 


8658 


8 7 3o 


8802 


72 


601 


8874 


8947 


9019 


9091 


9i63 


9236 


93o8 


9 38o 


9452 


9524 


72 


602 


9 5 9 6 


9 66 9 


974i 


9813 


9 885 


9957 


••29 


•101 


•i 7 3 


•245 


72 


6o3 


78o3i 7 


o38 9 


0461 


o533 


o6o5 


0677 


0749 
1468 


0821 


o8 9 3 


0965 


72 


604 


1037 


no 9 


1181 


1253 


1324 


1396 


1 540 


1612 


1684 


72 


6o5 


1755 


1827 


1899 


1971 


2042 


2114 


2186 


2258 


232 9 


2401 


72 


606 


2473 


2544 


2616 


2688 


2 7 5 9 


283 1 


2902 


2974 


3o46 


3ii7 


72 


607 


3i8 9 


3260 


3332 


34o3 


3475 


3546 


36i8 


368o 
44o3 


376l 


3832 


71 


608 


3904 


3 9 75 


4046 


4118 


4189 


4261 


4332 


4475 


4546 


7» 


609 


4617 


468 9 


4760 


483 1 


4 9 02 


4974 


5o45 


5u6 


5l87 


6259 


7« 


610 


78533o 


54oi 


5472 


5543 


56i5 


5686 


5 7 5 7 


5828 


58 99 


5970 


7i 


611 


6041 


6112 


6i83 


6254 


6325 


63 9 6 


6467 


6538 


66o 9 


6680 


7i 


612 


6 7 5i 


6822 


6893 


6 9 64 


7o35 


7106 


7H7 


7248 


7 3i 9 


73 9 o 


71 


6i3 


746o 


753 1 


7602 


■7673 


7744 


78i5 


7885 
85 9 3 


7 9 56 
8663 


8027 


8098 


7i 


614 


8168 


823 9 


83io 


838i 


845 1 


8522 


8734 


8804 


71 


6i5 


88 7 5 


8 9 46 


9016 


9087 


9157 


9228 


9299 


9369 


9440 


95io 


71 


616 


958i 


9 65i 


9722 


9792 


9863 


9933 


•••4 


••74 


*o 4 ^ 


•2l5 


70 


617 


790285 


o356 


0426 


0496 


0567 


0637 


0707 


0778 


08^8 


0918 


70 


618 


0988 


io5 9 


1 1 29 


1199 


1269 


1 340 


1410 


1480 


i55o 


1620 


70 


6vp 


1691 


1761 


i83i 


1901 


1971 


2041 


2 1 1 1 


2181 


2252 


2322 


70 


620 


792392 


2462 


2532 


2602 


2672 


2742 


2812 


2882 


2 9 52! 3022 


70 


621 


3092: 3i62 


323i 


33oi 


33 7 i 


3441 


35u 


358i 


365ij 3721 


70 


622 


3790! 386o 


3930 


4000 


4070 


4i3 9 


420 9 


4279 


434 9 44i8 


70 


623 


4488 


4558 


4627 


4697 


4767 


4836 


4qo6 


4976 


5o45 


5u5 


70 


624 


5i85 


5254 


5324 


53 9 3 


5463 


5532 


56o2 


5672 


574i 


58n 


70 


6 2 5 


588o 


5 9 4 9 


6019 


6088 


6i58 


6227 


62 9 7 


6366 


6436 


65o5 


69 


626 


65 7 4 


6644 


6713 


6782 6852 


6921 


6 99 o 


7060 


7129 


7.98 


69 


627 


7268 


7337 


74o6 


7475| 7545 


7614 


7683 


7 7 52 


7821 


7890 


69 


628 


7960 


8o2 9 


8098 


8167I 8236 


83o5 


83 7 4 


8443 


85i3 


8582 


69 


629 


865 1 


8720 


8789 
9478 


8858 8 9 27 


8996 


9065 


9 i34 


9203 


9272 


6 9 


63o 


799341 


9400 

oo 9 8 


9 547 9616 


9 685 


97 54 


9823 


9892 


9961 


69 


63 1 


800029 


0167 


o236i o3o5 


0373 


0442 


o5n 


Q380i 0648 


69 


632 


0717 0786 


o854 


O923I 99 2 


1061 


1 1 29 


1 1 98 1 266I i335 


69 


633 


1404 1472 


i54i 


1609! 1678 

22 9 5' 2363 


1747 


i8i5 


1884 I 9 52 2021 


69 


634 


2089 2 1 58 


2226 


2432 


25oo 


2568 2637! 2705 


a 


635 


2774 2842 


2910 


2979! 3o47 


3i 16 


3i84 


3252 332i 338g 


636 


3457 3525 


3o 9 4 


3662 373o 


3 7 o8 
4480 


386 7 


3935 


4oo3 4071 


68 


63 7 


4 1 3 9 4208 


4276 


43441 44i2 


4548 


4616 


4685; 4753 


68 


638 


4821 ( 4889 


49 5 7 


5o25 5oq3 


5i6i 


5229 
5908 


5297 


5365 5433 


68 


63 9 


55oi 5569 


5637 57o5|- 5773 


584i 


5976 


6044J 61 1 2 


68 


N. 


| 1 


* i 3 | 4 


5 


6 


7 


8 ] 9 


D j 





A TABLE 


OF LOGARITHMS FROM 1 


TO 


10,00a 


11 


>. 


1 « 


■ 1 > 1 3 


4 


5 j 6 


7 1 8 | 9 


D. 


64o 


806180 6248! 63i6 6384 


645i 


65ig 6587 


66551 6723; 6790 


68 


64i 


6858 6926! 69941 7061 


7129 


7'97j 7264 


7332| 7400 7467 


68 


642 


7535 7 6o3; 76701 7738 


7806 


78731 7941 


8008! 8076 


8143 


68 


643 


821 ll 8279: 8346| 8414 


•8481 


8549i 8616 


8684! 8 7 5 1 


8818! 67 


644 


8886 8953 


9021 9088 


9 1 56 


9223 9290 


9358) 9425 


9492 


67 


645 


g56o 9627 


96941 9762 


9829 


9896 9964 


••3 1 ••98 


•i65 


67 


646 


8ioa33 o3oo 


0367 0434 


o5oi 


0569 o636 


0703 0770 


08.37 


67 


647 


0904 0971 


1039 1 106 


1 173 


1240 


i3o7 


1374 I44i 


i5o8 


67 


643 


1373 1642 


1709 1776 


1843 


1910 


1977 


2044 


2111 


2178 67 


649 


2245 23l2 


2379! 2445 


25l2 


"79 


2646 


2713 


2780 


2847! 67 


65o 


8i2gi3; 2980 


3o47 3 1 14 


3i8i' 


3247 


33i4 


338i 


3448 


35i4 


67 


65i 


358 1! 3648 


3 7 i4 


3 7 8i 


3848 


3914 


3 9 8i 


4048 


4114 


4181 


67 


652 


4248 43 1 4 


438i 


444 7 


4314 


458 1 


4647 


4714 


4780 


4847 


67 


653 


49 1 3 4980 


5o46 


5u3 


5*79 


5246 


53 1 2 


53 7 8 


5445 


55n 


66 


654 


5578 5644 


57 1 1 


5777 


5843 


5910 


5 97 6 


6042 


6109 


6170 


66 


655 


6241! 63o8 


63 7 4 


644o 


65o6 


6573 


663g 


6705 


6771 


6838 


66 


656 


6904 : 6970 


7o36 7102 


7169 


7235 


73oi 


7 36 7 


7433 


7499 


66 


657 


7365j 7 63 1 


76981 7764 
83581 8424 


783o 


7896 
8556 


7962 
8622 


8028 


8094 


8160 


66 


658 


8226, 8292 
8885i 8951 


8490 


8688 


8754 


8820 


66 


659 


9017 9083 


9149 


92i5 


9281 


9346 


9412 


9478 


66 


66o 


819344 9610 


9676, 9741 


9807 


9S73! 9939 


••©4 


••70 


•i36 


66 


66i 


820201] 0267 


o333 0399 


0464 


o53o 


0095 


0661 


0727 


0792 


66 


662 


0838, 0924 


0989I 1035 


1120 


1 186 


I25l 


1317 


i382 


1448 


66 


663 


i5i4 1379 


1643 1710 


1773 


1841 


1906 


1972 


2o37 


2103 


65- 


664 


2168 2233 


2299! 2364 


243o 


2495 


256o 


2626 2691 


2756 


65 


665 


2822 2887 


2932 3oi8 


3o83 


3i48 


32i3 


3279 3344 
3g3o 3996 


3409 


65 


666 


3474 3539 


36o5j 3670 


3 7 35 


38oo 


3865 


4061 


65 


667 


4126 4191 


4256 4321 


4386 


445 1 


45i6 


458i 


4646 


47 II 


65 


668 


4776 4841 


4936; 4971 


5o36 


5ioi 


5i66 


5a3i 


5296 


536i 


65 


669 


5426 5491 


5556j 562i 


5686 


5701 


58 i 5 


5S8o 


5q45 


6010 


65 


670 


826073 6140 


62o4 ; 6269 


6334 


6J99 


6464 


6528 6d 9 3 
71731 7240 


6658 


65 


671 


6723 67S7 


6S52| 6917 


6981 


7046 


7111 


73o5 


65 


672 


7 36 9 7434 


7499 1 7563 


7628 


7692 


7737 


782 1 1 78S6 


79 5i 


65 


673 


8oi3| 80S0 


8144, 8209 


8273 


8338 


8402 


8467 853 1 


85g5 


64 


674 


8660 8724 


87S9 8853 


8918 


8982 


9046 


91 1 1 


9175 


9239 


64 


675 


93o4 9368 


9432 9497 


9 56i 


9625 9690 


97 54 


9818 


98S2 


64 


676 


9947 0# ii 


••75 | «i3 9 


•204 


®268 «332 


•3g6 


•460 


•525 


64 


677 


83o589; o653 


0717 1 07S1 


o845 


0909 0973. 


1037 


1102 


1 166 


64 


678 


i23oj 1294 


i358, 1422 


i486 


i55o| 1614 


1678 


1742 


1806 


64 


679 


1870, 1934 


I998 t 2062 


2126 


2189 2253 
2828 2892 


2317 


233i 


2445 


64 


680 


832009; 2373 


2637; 2700 


2764 


2g56 


3020 


3o83 


64 


681 


3147 32i 1 


3275 3338 


3402 


3466| 353o 


35 9 3 


365 7 


3721 


64 


682 


3 7 84 3S48 


3912' 3973 


4o3 9 


4io3 4106 


423o 


4294 


4357 


64 


683 


4421, 4484 


4348: 461 1 


4675 


4739 4802 


4866 


4929 


4993 


64 


684 


5o56| 5i20 


5i83| 5247 


53io 


53 7 3 


5437 


55oo 5564 


5627 


63 


685 


Ougi 1 5734 


5817 588i 


5944 


6007 


6071 


6i34| 6197, 6261 


63 


680 


6324 6337 


6431! 65i4 


6077 


6641 


6704 


6767 683o 6894 
73991 7462; 7525 


63 


687 


6937 7020 


7o33 ! 7146 


7210 


7273 


7 336 


63 


688 


7088 7632 


77i 5 ! 7778 


7841 


7904 


7967 


8o3o, 8093 81 56 


63 


6«9 


8219" 8282 


8345: 8408 


8471 


8334 


8397 


8660 8723 8786 


63 


690 


838849 8912 


8973 9 o38 


9101 


9164 9*27 


9289 9352 9413 


63 


691 


9478 9341 9604 9667 


9729 


9792 


9833 


9918 9981 »®43 


63 


J 9 ? 


840106 0169 0232 0294 


o357 


0420 


0482 


o545 0608 0671 


63 


693 


0733, 0796; 0869 0921 


0984 


1046 


1 109 


1172 1234 1297 


63 


694 


i339 ! 1422 1483 i547 


1610 


1672 


1735 


1797 i860 1922 


63 


693 


I985j 20471 2110 2172 


2235 2297I 236o 


242 2 ! 2484 23i-7 


62 


696 


2609 2672; 2734 2796 


2859 292 1 1 2983 


3o46 3io8 3170 


62 


697 


3233 3293 3357 3420 


3482 3544 36o6 


3669 373: 3793 


62 


698 


3855 3918 3980 4042 


4104 4«66! 4229 


42911 /,353 44i 5 


62 


699 


4477, 4539 


4601 4664' 4726 4788! 4830 


4912 4974 5o36 


62 


H. 


I 1 


2 3 ' 4 ! 5 j 6 


"7" i"l~i~ 


- j 



1] 



12 



A TAIiLE OF LOGARITHMS FROM 1 TO 10,000. 



N. | 


| , 


2 


3 ! 


4 j 5 | 6 | 7 | 8 


9 1 


*>l 


700 


845098: 5l6o 52221 5284 


5346 54o8l 5470, 5532' 5594! 


56561 62 


701 


57i8j 5780: 5842 i 5904 


5966 6028I 6090 6 1 5 1 1 62 1 3| 6275! 62 


"!02 


633]; 6399! 646 if 6523 


6585 6646 


6708 6770! 6832 68941 62 


to3 


6955! 7017 


7079 7141 


7202' 7264I 


7326I 7 388' 7449 75 n 


62 


704 


7573; 7 634 


7696 7708 


7819! 7881 


79431 8004S 8066 8128 


62 


70D 


8189; 825i 


83 1 2 8374. 


8435' 


84 9 7' 


8559 8620 8682I 8743 


62 


706 


88o5 8866 


8928 8989' 


905 1 


91 12 


9174-1 9235j 9297 9358! 61 


707 


9419 9481 


9542 9604 


9 665 


9726 


9788! 9849! 991 1 1 9972' 61 


708 


85oo33 0095 


oi56> 0217 


0279 


o34o 


040 1 


0462 


o?24 o58d 


61 


709 


0646 0707 


0769 


o83o 


0891 


0952 


1014 


1075 


ii36 


1 197 


61 


710 


85i258 1320 


i38i 


1442 


i5e3 


1 564 


i625 


1686 


H47 


1809 


61 


711 


1870] 1931 


1992 


2o53 


2114 


2175 


2236 


2297 


2358 


2419 


61 


712 


2480! 2541 


2602 


2663 


2724 


2 7 85 


2846 


2907 


2968 


3029 


61 


7*3 


3090 : 3i5o 


321 1 


3272 


3333 


33 9 4 


3455 


35i6 


3377 
4i85 


3637 


61 


7i4 


36981 3759 


3820 


388i 


3g4i 


4002 


4o63 


4124 


4245 


6c 


7 i5 


4306 4367 


4428 


4488 


4549 


4610 


4670 ! 4 7 3i 


4792 


4852 


61 


716 


491 3 1 4974 


5o34 


5095 


5 1 56 


52i6 


5277 533 7 


53 9 8 


5459 


6t 


7H 


5519J 558o 


564o 


5701 


5 7 6i 


5822 


5882 j 5 9 43 


6oo3 


6064 


6t 


718 


6i24| 6i85 


6245 


63o6 


6366 


6427 


6487 6348 


6608 


66681 60 


719 


6729I 67S9 


685o 


6910 


6970 


7o3i 


709I 7ID2 


7212 


7272 


60 


720 


85 7 332 


7 3 9 3 


7453 


75i3 


7^74 


7634 


7694, 7755 


78i5 


7875 


60 


721 


$ 


79 9 5 


8o56 


8116 


8176 


8236 


8297 835 7 


8417 


8477 


60 


722 


85 97 


865 7 


8718 


8778 


8838 


8898 


8 9 58 


9018 


9078 


60 


7 23 


9 i38 


9 , 9 8 


9 258 


93 1 8 


9 3 7 o 
9978 


9439 


9499 


9 55 9 


9619 


9679 


60 


724 


n ,Wl% 


9799 


9 85 9 


9918 


«»38 


©» 9 8 


•i58 


•218 


•278 


60 


7 25 


86o338 


o3 9 8 


0458 


o5i8 


0578 


0637 


0697 


0757 


0817 


0877 


60 


726 


0937 


0996 
1 5g4 


io56 


1116 


1 176 


1236 


1295 


i355 


Ui5 


1475 60 


727 


1 534 


i65 4 


1714 


1773 


1 833 


i8 9 3 


1952 


2012 


2072 


60 


728 


2l3l 


2191 


225l 


23lO 


2370 


243o 


2489 


2549 


2608 


2668 


60 


729 


2728 


2787 


2847 


2906 


2966 


3o25 


3o85 


3i44 


3204 


3263 


60 


73o 


863323 


3382 


3442 


35oi 


356i 


3620 


368o 


3739 


3799 


3858 


5 9 


7 3i 


3 9 n 


3o77 
4570 


4o36 


4096 


41 55 


4214 


4274 


4333 


4392 


4452 


39 


7 32 


45 11 


463o 


4689 


4748 


4808 


4867 


4926 


4985 


5o',5 


5 9 


7 33 


5io4| 5i63 


5222 


52S2 


5341 


54oo 


5459 


5519 


5578 


5637 


5 9 


734 


5696 5755 


58i4 


58 7 4 


5 9 33 


5 99 2 


6o5i 


61 10 


6169 


62^8 


5 9 


735 


6287I 6346 


64o5 


6465 


6524 


6583 


' 6642 


6701 


6760 


6819 


5 9 


736 


6878 6937 


6996 


7o55 


71U 


7173 


7232 


7291 


735o 


7409 


^ 9 


737 


74671 7526 


7385 


7644 


7703 


7762 


7821 


7880 


7939 


799 s 


5 9 


738 


8o56 81 15 


8174 


8233 


8292 


835o 


8409 8468 


8527 


8586 


5 9 


7 3 9 


8644 | 8703 


8762 
9349 


8821 


8879 


8 9 38 


8997 9056 


9114 


9173 


5 9 


740 


8692321 9290 


9408 


9466 


9525 


9584 


9642 


9701 


9760 39 


74i 


9818 


9877 


993 d 


9994 


•°53 


•in 


o I7 o 


•228 


•287 


©345 59 
0930 58 


742 


870404 


0462 


0521 


0379 


o638 


0696 


0755 


oSi3 


0872 


743 


0989 
1073 


1047 


1 106 


1 164 


1223 


-1281 


1339 


i3 9 8 
1981 


1456 


i5i5| 53 


744 


i63i 


1690 


1748 


1806 


i865 


i 9 23 


2040 


2098 58 
2681 53 


745 


2i56 


22l5 


2273 

2855 


233 1 


238g 


2448 


25o6 2564 


2622 


746 


2739 2797 


2913 


2972 


3o3o 


3o88| 3146 


"3204 3262I 58 


747 
748 


332i 1 33 79 3437 


349 5 


3553 


36n 


3669J 2727 


3785 


3844! 58 


3902J 3960] 4018 


4076 


4i34 


4192 


425o 4308 


4366 


4424 ; 53 


74g 


4482 4540! 4598 


4656 


47i4 


4772 


483o 4888 


4945 


5oo3i 53 


730 


875061 5119 1 5177 


5235 


5293 


535i 


5409 j 5466 


5524 


5582! 53 


7 5i 


5640 


5698 5756 


^8i3 


58 7 i 


5g2g 


3987 6o45 
6564 6622 


6102 


6160 


58 


702 


6218 


6276! 6333 


63 9 i 


6449 


65o7 


6680 


6 7 3 7 


58 


753 


6795 


6853 6910 


6968 


7026! 70S3 


7141 7199 


7256 


73i4 


58 


754 


7 3 7 i 


7429 74871 7344 


7602 7639 7717J 7774 


7832 


7889 
8464 


58 


755 


7947 


8004: 8062! 81 19 


8177 8234! 8292J 8349 


8407 


57 


7 56 


85221 8579 ; 8637; 8694 


8732 8809 8866 


8924 


89S1 


9039 


^ 7 


7 5 7 


9096! 9i53 921 1 


9268 


9325 9383 9440 


9497 


9555 


9612 


57 


758 


9609, 9726 1 9784 


9841 


989S 9906 1 «®i3 
0471 o528 o585 


••70 


•127 


•i85 


I 1 


75 9 


880242 0299 o356 ; o4i3 


0642 


0699 


0756 


37 


N. 


| 1 j 2 | 3 


4 | 5 | 6 


7 j 8 


9 





A TABLE 


OF . 


LOGARITHMS FROM 1 


TO 


10,000 


12 


N. 





1 


2 


_J_ 


4 


5 


6 | 7 | 8 


9 


1). 


1 76o 


880814 


0871 


0928 


0985 


1042I 1099 


n56l i2i3j 1271 


"i328 


57 


1 761 


i385 


1442 


1499 


1 556 


i6i3 1670 


1727 


17841 l84l 


1898 


5 7 


762 


1 905 


2012 


2069 


2126 


2 1 83 2240 


2297 


2354 241 1 


2468 


57 


703 


2525 


258i 


2038 2695 


2752! 2809 


2866 


2923; 2980 


3o37 


57 


764 


3093 


3i5o 


3207 


3264 


332i 3377 


3434 


3491 3548 


36o5 


57 


765 


366 1 


3 7 i8 


3775 


3832 


3888 


3945 


4002 


4059 41 15 


4172 


57 


766 


4229 


4285 


4342 


4399 


4455 


45 1 2 


4D69 


4625 


4682 


4739 
53o5 


2 7 


767 


4795 


4852 


4909 


4965 


5022 


5o 7 8 


5i35 


6192 


5248 


57 


768 


536 1 


5418 


5474 


553 1 


5587 


5644 


5700 


5 7 5 7 


58i3| 58 7 o 


57 


769 


5926 


5 9 83 


6039 


6096 


6i52 


6209 


6265 


632i 


63 7 8; 6434 


56 


770 


886491 
7054 


6547 


6604 


6660 


6716 


6 77 3 


6829 


6885 


6942 


6998 


56 


77i 


7111 


7167 


7223 


7280 


H 36 o 


7392 


7449 


73o5 


756i 


56 


772 


7617 


7674 


7730 


7786 


7842 


7898 


79 55 


8011 


8067 


8i23 


56 


773 


8179 


8236 


8292 


8348 


8404 


8460 


85i6 


85 7 3 


8629 


8685 


56 


774 


8741; 8797 
9 302| 9358 


8853 


8909 


8 9 65 


9021 


9077 


9134 


9190 


9246 


56 


770 


9414 


9470 


9626 


9 582 


9 o3 8 


9694 


9730 


9806 


56 


776 


9862 1 9918 


9974 


•«3o 


••86 


•141 


•'97 


•253 


•3o 9 


•365 


56 


IT 


890421 1 0477 


o533 


o589 


0645 


0700 


0736 


0812 


0868 


0924 


56 


778 


0980 


1033 


1 09 1 


1 147 


1203 


1259 


i3U 


1370 


1426 


1482 


56 


779 
780 


1537 


1593 


1649 


l7o5 


1760 


1816 


1872 


1928 


l 9 83 


2039 


56 


892095 


2i5o 


2206 


2262 


23i7 


2373 


2429 


2484 


254o 


2595 


56 


78. 


?65i 


2707 


2762 


2818 


2873 


2929 


2085 1 3o4o 
35401 3595 


3096 


3i5i 


56 


782 


3207 


3262 


33i8 


33 7 3 


3429 


3484 


365 1 


3706 


56 


783 


3762 


3817 


38 7 3 


3928 


3o84 


4o39 


4094 4<'5o 


42o5 


4261 


55 


784 


43 16 


4371 


4427 


4482 


4^38 


4593 


4648 4704 


4759 


48:4 


55 


785 


4870 


4925 


4980 


5o36 


509 1 


5i46 


520I| 5257 


53i2 


5367 


55 


786 


5423 


5478 


5533 


5588 


5644 5699 


5764! 5809 


5864 


5920 


55 


787 


5975 


6o3o 


6o35 


6140 


6195) 62D1 


63o6 636i 


6416 


6471 


55 


788 


6D26 


658i 


6636 


6692 


67471 6802 


68 37 6912 


6967 


7022 


55 


789 


7077 


7132 


7187 


7^42 


72971 7352 


7407 7462 


7317 


7 5 7 2 
8122 


55 


790 


897627 


7602 


7737 


7792 


7847I 79 02 


7907! 8012 


8067 


55 


791 


8176 


8i3 1 


8286 


8341 


83 9 6 845i 


85o6! 856i 


86i5 


8670 


55 


792 


8725 


8780 


8835 


8890 


8944 8999 


9o5i 9109 


9164 


9218 


55 


7 9 3 


9273 


9 3a8 


9383 


9437 


9492 9 5 47 


9602 1 9636 


..97 1 1 


9766 


55 


79^ 


9^21 


9 8 7 5 


99 3 ° 


99SJ 


••39 «»94 


•i4q «203 


•258 


•3l2 


55 


79D 


900367 


0422 


0476 


o33i 


o586 


0640 


06961 0749 


0804 


0859 


55 


796 


0913 


09681 1022 


1077 


ii3i 


1 186 


1240! 1295 


1349 


1404 


55 


797 


1458 i5i3 1567 


1622 


1676 


i 7 .3i 


1735 1840 


1894 


1948 


54 


798 


2oo3j 2o57j 21 12 


2166 


2221 


2270 


232qI 2384 


2438 


2492 


54 


799 


2547) 2601 2655 


2710 


2764 


2818 


28 7 3j 2927 


2981 


3o36 


54 


800 


9030901 3i44i 3199 


3253 


3307 


336i 


34i6| 3470 


3324 


3378 


54 


801 


36331 3637 3 7 4i 


3795 


3849 


3904 


3g58| 4012 4066 


4120 


54 


802 


4174 4229 42S3 


4337 


4391 1 4443 


4499 4553! 4607 
5o4o 5094 5u8 


4661 


54 


8o3 


4716 477° 4824 


4878 


4932 4986 


5202 


54 


804 


5256 53 10 5364 


54.8 


5472I 5526 


558o 5634| 5633, 5742 


54 


8o5 


5796 585oj 5904 


5 9 58 


6012' 6066 


61 19 6 1 73 j 6227 


628l 


54 


806 


6335 


638 9 6443 


6497 


655i 


6604 


6658, 6712! 6766 


6820 


54 


807 


6874 


6927I 6981 


7o35 


7089 


7143 


7196I 7250 73o4 


7 358 


54 


808 


741 1 


7465J 7519 


7 5 7 3 


7626 


7680 


7734, 7787 7841 7893 


54 


809 


7949 8002 8o56 


8110 


8i63 


82.7 


82701 8324! 83 7 8! 843i 


54 


810 


90S485J 8539 8592 


8646 


8699 


8 7 53 


88ot 836o, 8914I 8967 


54 


811 


9021! 9074 9128 


9181 


9 233| 92S9 


93421 g3o6 94491 93o3 


54 


812 

8i3 


9556J 9610 9663 
910091 0144 0197 


9716 

025l 


9770; 9823: 98771 99 3o 99841 ••37 
o3o4i o358: 0411, 0464 o5i8| 0571 


53 
53 


814 


06241 0678 0731 


0784 


o838 0891 0944 0998 


io5ii tio4 


53 


8i5 


n58: 1211 1264 


i3 1 7 


1371 1424 1 4-77J i53o 


1 584 1637 


53 


81S 


1690 1743 1797 


i85o 


1903 1906 20091 2o63 


2116 2169 


53 


817 


2222! 2275 2328 


238i 


2435! 2488 2341 i 2594 


2647 J 2700 


53 


818 


2753j 2806 2859 


2913 


sg66 3019 3072I 3i 25 


3178I 323i 


53 


819 


32841 3337 33 9 


3443 


3496I 3549 36oal 3655 


3708J 376! 


53 


"n. 


1 ;, | 2 


3 | 4 | 5 | 6 | 7 


~8 | 9 


~D- 



4 


A TABLE OF LOGARITHMS FROM 1 


TO 


10,00 


0. 




N. 





, 1 , j 


3 | 4 | 5 | 


6 j 7 


8 1 9 I D. 


820 


9 i38i4 


3867I 3 9 20 


3973 


40261 4079 


4i32| 4184 


4237I 4290 53 


821 


4343 


4396' 4449I 


45o2 


4555! 4608 


4660 47 t 3 


4766: 4819! *3 


822 


4872 


4925 4977 


5o3o 


5o83i 5 1 36 


5189! 5241 


5294 5347 53 


823 


5400 


5453: 55o5 


5558 


56u| 5664 


5716! 5769 


5822! 5875 


53 


824 


5927 


5980 6o33 


6o85 


6i38i 6191 


6243 6296 


6349' °4°i 


53 


823 


6454 


65o7 6559 


6612 


6664 6717 


6770 


6822 


6875 6927 


53 


826 


6980 


7o33 7085 


7i38 


7190 


7243 


7293 


7348 


74oo| 7453 


53 


827 


7306 

8o3o 


7 558 761 1 


7663 


7716 


7768 


7820 


7873 


7925. 7978 


52 


828 


8o83 8 1 35 


8188 


8240 


82 Q 3 


8345 


83 Q7 


845o! 85o2 


52 


829 


8555 


8607 8659 


8712 


8764 8816 


8869 


8921 


8973! 9026 


52 


83o 


919078 


9i3o 9183 


9235 


9287 9340 


9392 9444 


9496: 9549 


52 


83 1 


9601 


9653 9706 


9758 


9R10; 9R62 


9914 


9967 


••191 "71 


52 


832 


920123 


0176 0228 


0280 


o33a| o384 


0436 


0489 


o54i 


o5g3 


52 


833 


o645 


0697, 0749 


0801 


o853i 0906 


0908 


1010 


1062 


11 14 


52 


834 


1166 


1218 1270 


1322 


1374! 1426 


1478 


i53o 


i582 


1634 


52 


835 


1686 


1738 1790 


1842 


1894 1946 


1998 


2o5o 


2102 


2i54 


52 


836" 


2206 


2258 23lO 


2362 


2414 2466 


25i8 


2570 


2622 


2674 


52 


837 
838 


2725 


27771 2829 


2881 


2933I 2985 


3o3 7 


30S9 


3 1 40| 3192 


52 


3244 


3296I 3348 


3399 


345i| 35o3 


3555 3607 


3658 3710 


52 


83 9 


3762 


3814 3865 


3917 


3969 4021 


4072 4124 


4176 4228 


52 


840 


924279 


433i 4383 


4434 


4486| 4538 
5oo3i 5o54 


4589j 4641 


46 Q 3 4744 


52 


841 


4796 


4848, 4899 
5364 54i5 


4951 


5io6: 5i57 


5209j 5261 


52 


842 


53i2 


5467 


5 5 1 8 : 5570 


562 1 i 5673 


5725 5776 


52 


843 


5828 


5879 5 9 3i 


5982 


6o34' 6o85 


6137; 6188 


6240 6291 


5i 


844 


6342 


63g4 6445 


6497 


6548 ; 6600 


665 1 1 6702 


6754 68o5 


5i 


845 


6857 


6908 6959 


701 1 


7062 


71 14 


7i65 7216 


7268! 7319 


5i 


846 


7370 


7422! 7473 


7524 


7 5 7 6 


7627 


7678; 77 3o 


7781' 7832 


5i 


848 


7883 


7 q35, 7986 


8o3; 


808S 


8140 


9191: 8242 
8 7 o3 8 7 54 


82 9 3| 8345 


5i 


83 9 6 


8447! 8498 


8549 


8601 


8652 


88o5: 8857 


5 1 


849 


8908 


8959 9010 


9061 


9112 1 9163 


g2i5 9266 


9I17 1 9368 


5i 


85o 


929419 


9470 9'' )21 


9 5 7 2 


9623 9674 


9725; 9776 


9827: 9879 


5i 


85 1 


993o 


9981! ««32 


•®83 


•i34i *i85 


®236i 0287 


•338; «38 9 
0847! 0898 


5i 


852 


93o44o 


0491 o542 


0592 


0643 0694 


0745 0796 


5i 


853 


0949 


1000 io5i 


1 102 


1 1 53T 1204 
16611 1712 


1254 i3o5 


1356' 1407 


5i 


854 


1458 


i5o9 i56o 


1610 


i763| 1814 


i865j 1916 


5i 


855 


1966 


2017 2068 


2118 


2169J 2220 


227I 2322 


2372 2423 


5i 


856 


2474 
2981 


2524 2575 


2626 


2677 2727 


2778! 2829 


2879^.2930 


5i 


85 7 


3o3ii 3o82 


3i33 


3i83| 3234 


3285; 3335 


3386; 3437 


5i 


858 


3487 


3538| 358g 


3639 


36 9 o 3740I 3791 384i 


38 9 2l 3 9 43 


5i 


85 9 


3993 


4044; 4094 


4i45 


4ig5 4246J 4296! 4347 


43 9 7 


4448 


5i 


860 


934498 


4549 4599 


465o 


4700J 4751 1 4801 4852 


4902 


4g53 


5o 


861 


5oo3 


5o54 5io4 


5.54 


52o5j 5255 53o6 5356 


5406 


5457 


5o 


862 


55o7 


5558, 56oS 


5658 


57091 5759) 5809' 586o 


5910 


5960 


5o 


863 


6on' 6061 61 1 1 


6162 


62121 62621 63i3 6363 


64i3 


6463 


5o 


864 


65i4 


6564; 6614 


6665 


671 5| 6765 


68 1 5, 6865 


6916 


6966 


5o 


865 


7016 


7066! 71 17 


7167 


7217! 7267 


7317 7367 


74i8 


7468 


5o 


866 


75i8 


7 568 7618 


7668 


77181 7769 


7819 7869 


7919 


7969 


5o 


867 


8019 


8069! 8119 


8169 


8219 8269 


8320 


8370 


8420J 8470 


5o 


868 


8520 


8570 8620 


8670 


8720 


8770 


8820 


8870 


89201 8970 


5o 


869 


9020 


9070 9120 


9170 


9220 


9270 


9320 


9369 


9419 9469 


5o 


870 


939519 


9569; 9619 


9669 


9719 
0218 


9769 9S19 


9869 


991 8| 9968 


5o 


871 


9400 1 8 


0068; 01 18 


0168 


0267 


o3i7 


o367 


0417 


0467 


5o 


872 


o5i6 


o566 : 0616 


0666 


0716 


0765 


081 5 


o865 


0915 


0964 


5o 


8 7 3 


1014 


1064 1 1 1 4 


n63 


I2l3 


1263 


i3i3 


1362 


1412 


1462 


5o 


874 


1 5i 1 


i56i 1611 


1660 


1710 


1760 


1809 


i85 9 


1909 195^ 


5o 


87J 


200S 


2o58i 2107 


2157 


2207 


2256 23o6 


2355 


24o5j 2455 


5o 


876 


2D04 


2554 26o3 


2653 


2702 


2752J 2801 


285i 


2QOI | 2950 


5o 


*77 


3ooo 


| 3o4q ] 3oqq 


3i48 


3198! 3247 3297 


3346 


33g6 3445 


5 9 


878 


3493 1 3544' 35g3 


3643 


3692; 3742 3791 


384i 


38qo 3g3g 


^9 


B79 


3989 4o38 40S8 


i 4137 


4186 4236| 4285! 4335 4384! 4433 


5 9 


N. 


| I 1 2 


L» 


4 j 5 | 6 ) 7 1 8 | 9 j D. 



A TABLE OF LOGARITHMS FROM 1 TO 10,000. 



15 



N. 


| , 


2 


3 | 4 


5 | 6 


7 


8 1 9 


D. 


88o 


944483 4532 


458i 


463 1 


4680 1 4729! 4779: 


48281 4877; 4927 


49 


88i 


4976. 5o25 


5074 


5i24 


5l73 5222 ! 5272; 


532i 5370 54191 49 


882 


5469 1 55i8 5567 


56i6 5665 67 1 5| 5764! 


58i3| 5862 5912 


49 


883 


5961 6010 


6039 


6io8 : 61D7: 6207 6256; 


63o5| 6354 64o3 


49 


884 


6402 : 65oi 


655i 


6600 6649' 66981 6747 ] 


6796! 6845 6894 


49 


885 


69431 6992 


7041 


709o ; 7140; 7189 7238 


7287: 7336 7 385 


49 


886 


7434: 7483 


7532 t58i I 763oi 7679' 7728 


7777' 7826 7873 


49 


887 


7924 7073 


8022 8070' 8iiq' 8168 8217 


8266 83 1 5 8364 


49 


888 


84i3; 


8462 


85n 


856o' 86091 8637 


8706: 


8 7 55| 8804 8853 


49 


889 


8902 


8931 


8999 


9048 9097 


9146 


9i95 ! 


9244 9292 9341 


49 


890 


949390 


9439 


9488 


9336 9383 


9634 


9 683 ' 


9731 


9780; 9829 


49 


891- 


9^78 


9926 


9973 


•®24 *°73 


®12I 


©170 


•219 


•267; »3i6 49 


89? 


9 5o365 


0414 


0462 


o5i 1 1 o56o 


0608 


0657 


0706 


0754; o8o3 


49 


893 


oS5i 


0900 


0949 


0997 1046 


1095 


Ii43 


1192 


1240I 1289 


49 


894 


1338 


i386 


1435 


1483, i532 


i58o 


1629 


1677 


1726 1775 


49 


89 5 


1823 


1872 


1920 


1969 


2017 


2066 21 1 4 


2i63 


22m 2260 


48 


896 


23oS 


2356 


24o5 


2453 


25o2 


255o! 2599 


2647 


2696J 2744 


48 


897 


2792 


2841 


2889 2938 


2986 


3o34 3o83 


3i3i 


3i8o, 3228 


48 


898 


32 7 6 


3325 


33 7 3 3421 


3470! 35i8 


3566 


36i5 


3663! 37 1 1 


48 


899 


3760 


38o8 


38561 3905 


3953 


4001 


4049 


4098 


4146! 4194 


48 


900 


954243 


4291 


433 9 438 7 


4435 


4484 


4532 


458o 


4628! 4677 


48 


901 


4725 


4773 


4821 


4869 


4918 


4966 5oi4 


5o62 


5no| 5i58 


48 


902 


5207 


5235 


53o3 


535i 


53 9 9 

588o 


5447 5495 
5 9 28 5976 


5543 


5G92 5640 


48 


903 


5688 


5736 


5 7 84 


583 2 


6024 


6072 6120 


48 


904 


6168 


6216 


6265 


63i3 


636i 


6409 6457 


65o5 


6553 6601 


48 


903 


6649 
7128 


6697 


6745 


6793 


6S40 


6888 6 9 36 


6984 


7032! 7080 


48 


906 " 


7176 


7224 


7272 


7320 


73681 7416 


7464 


7512; 7559 


48 


907 


7607 
. 8086 


7 655 


7703 


775i 


7799 


7847 7894 


7942 


7990' 8o38 


48 


908 


8i34i 8181 


8229 


8277! 8325! 8373 


8421 


8468 85 16 


48 


909 


8564 


8612 


865g 


8707 


8755, 88o3: 885o 


8898 


8946, 8994 


48 


910 


959041 


9089 


9137 


9185 


9232 9280 9328 


9 3 7 5 


9423; 9471 


48 


911 


9318 


9 566 


9614 


9661 


9709 9757 9804 


9852 


9900! 9947 


48 


912 


9 99 5 


••42 


••90 «i38 


•i85| »233| *28o 


•328 


•376; «423 


48 


9 i3 


960.471 


o5i8 


o566 06 1 3 


0661 


07091 0756 


0804 


o85i] 0899 


48 


914 


0946 


0994 


10411 1089 


n36 


1184 I23l 


1279 


i326 


i3 7 4 


47 


giD 


1421 


1469 


i5i6| i563 


161 1 


i658 


1706 


i 7 53 


1801 


1848 


47 


916 


i8 9 5 


1943 


1990 2o38 


2o85 


2l32 


2180 


2227 


2275 


2322 


47 


917 


236 9 


2417 


2464 


25ll 


2359 


2606 


2653 


2701 


2748 


2795 


47 


918 


2843 


2890 


2937 


2983 


3o32 


3079 


3i26 


3i74 


3221 


3268 


47 


919 


33i6 


3363 


34io 


3457 


35o4 


3552 


3599 


3646 


36 9 3 


3741 


47 


920 


963788 


3835 


3882 


3929 


3977 


4024 


4071 


4118 


4i65 


4212 


47 


921 


4260 


43o7 


4354 44oi 


4448 


449 5 4542 


4590 


4637 


4684 


47 


922 


473 1 


4778 


4825 4872 


4919 


49661 5oi3 


5o6i 


5io8 


5i55 


47 


923 


5202 


5249 


5296 5343 


5390 


5437 5484 


553i 


55 7 8 


5625 


47 


924 


5672 


5719 


5 7 66 58 1 3 


586o 


5907 
63 7 6 


5 9 54 


6001 


6048 


6095 


47 


9 2D 


6l42 


6189 


6236 6283 


632 9 


6423 


6470 


65 1 7 


6564 


47 


926 


66l I 


6658 


6705 6752 


6799 6845 


6892 


6939 


6986 


7o33 


47 


9 "l 


7080 


7127 


7173 7220 


7267 7314 


736i 


74o8 


7454 


75oi 


47 


928 


7548 


7595 


76421 7688 


7 7 35i 7782 
82o3 8249 


7829 


7875 


7922 


7969 


47 


929 


8016 


8o62| 8109! 8i56 


8296 


8343 


83go 


8436 


47 


9 3o 


968483 


853o! 8376; 8623 


8670 1 8716 


8763 


S810 


8856 


8903 


47 


9 3r 


8950 


8996 


9043 9090 


9i36 9183 


9229 


9276 


o323 


9369 


47 


932 


94l6 


9 463 


9509J 9556 


9602 9649 


9 6 9 5 


9742 


9789 


9 835 


47 


933 


9882 


9928 


9973 ••21 


••68| »u4 


•161 


•207 


•234 


•3oo 


47 


9 34 


970347 


o3g3 


0440 j 0486 


o533 


0379 


0626 


0672 


O719 


0763 


46 


9 ^ 


o8l2 


o858 


0904; 0931! 0997 


1044 


1090 


u3 7 


n83 


1229 


46 


93 6 


I276 


l322 


1369J i4i5| 1461 


i5o8 


i554 


1601 


1647 


169: 


46 


9 ll 


1740 


I786 


i832 1879! 1925 


1971 


2018 


2064 


2110 


2107 


46 


938 


2203 


2249 


! 22 9 5 2342 2388 


2434 


2481 


2527 


25 7 3 


26 rn 


46 


989 


2666 


! 2712 


27581 2804! 2S5i 


2897 


2943 


2989 


3o35 3082 


46 


N. 


' 


1 1 


1 2 j 3 j 4 


5 ( 6 


7 


8 9 


D. 



16 


A TABLE 


OF 


LOGARITHMS FROM j 


TO 


10,000. 




N. 





1 


2 


3 I 4 | 5 j 


6 


7 


8 


9 


D. 
46 


940 


973128 


3i74 


3220 


3266 33 1 3; 335g> 


34o5 


345 1 


3497 


3543 


94i 


3590 


3636 


3682 


3728! 3 77 4 


3820 


3866 


3 9 i3 


3959 


4oo5 


46 


942 


4031 


4097 


4143 


4189 4235 


4281 


4327 


43 7 4 


4420 


4466 


46 


943 


4312 


45d8 


4604 


465q 4696 


4742 


4788 


4834 


4880 


4926 


46 


944 


4972 


5oi8 


5o64 


5i 10 


5i56 


5202 


5248 


5294 


5340 


5386 


46 


945 


5432 


5478 


5524 


5370 


56i6 


5662 


5707 


5 7 53 


5799 


5845 


46 


946 


589I 


5 9 3 7 


5 9 83 


6029 


6075 


6121 


6167 


6212 


6258 


63o4 


46 


947 


63ao 


63 9 6 


6442 


6488 


6533 


6579 


6625 


6671 


6717 


6763 


46 


948 


6808 


6854 


6900 


6946 


6992 


7o3 7 


7083 


7129 


7175 


7220 


46 


949 


7266 


7312 


7 358 


74o3 


7449 


74g5 


754i 


7586 


7632 


7678 
8i35 


46 


qdo 


977724 


7769 


78.5 


7861 


7906 
8363 


7932 


7998 
8454 


8o43 


8089 


46 


9 5i 


8181 


8226 


8272 


83i 7 


8409 


85oo 


8546 


85 9 i 


46 


952 


8637 
9093 


8683 


8728 


8774 


8819 


8865 


891 1 


8 9 56 


9002 


9047 


46 


9 53 


9i38 


9184 


9230 


9275 9321 


9366 


9412 


9457 


93o3 


46 


954 


9348 


9394 


9 63 9 


9685 


97 3o ! 9776 


9821 


9867 


99! 2 


9953 


46 


955 


980003 


0049 0094 


0140 


oiSd o23i 


0276 0322 


0367 


0412 


45 


966 


0438 


o5o3| o349 


0594 


064c o685 


0730 O776 


0821 


0867 


45 


9 5 7 


0912 


0957 ioo3 


1048 


1093 1 139 


Il84| 1229 


1275 


l320 


45 


9 58 


1 366 


1411 


1456 


i5oi 


1 547 1592 


i63 7 16831 1728 


1773 


45 


9 5 9 


1819 


1864 


1909 
2 362 


1954 


2000, 2043 


2090! 2i35 2181 


2226 


45 


960 


982271 


23i6 


2407 


2452 2497 


2543; 2588 


2633 


2678 


4.5 


961 


2723 


2769 


2814 


2S59 


2904 2949 


2994 3o4o 


3o85 


3i3o 


45 


962 


3173 


3220 


3265 


33io 


3356 3401 


3446 


3491 


3536 


358i 


45 


9 63 


3626 


3671 


3716 


3762 


3807 3852 


38 97 


3942 


3987 


4o32 


45 


964 


4077 


4122 


4167 


4212 


4257 43o2 


4347 


4392 


4437 


4482 


45 


965 


4327 


4572 


4617 


4662 


4707 4752 


4797 


4842 


4887 


4932 


45 


966 


4977 


5022 


6067 


5i 12 


5i57i 5202 


5247 


5292 


533 7 


5382 


45 


967 


5426 


5471 


55i6 


556i 


56o6 565i 


56 9 6 


5741 


5 7 86 


583o 


45 


968 


5873 


592O 

636 9 


6965 


6010 


6o55 6100 


6144 6189 


6234 


6279 


45 


969 


6324 


64i3 


6458 


65o3; 6548 


65g3 6637 


6682 


6727 


45 


970 


986773 


6S17 


6861 


6906 


6951 6996 
7398 7443 


7040 


7o85 


7i3o 


7. 7 5 


45 


971 


7219 


7264 


7 3 ?9 


7353 


7488 


7332 


7577 


7622 


40 


972 


7666 
8u3 


77" 


7736 


7800 


7845 7890 


79 34 


7979 


8024 


8068 


43 


973 


8157 


8202 


8247 


8291 


8336 


838i 


8423 


8470 


85i4 


45 


974 


855o 
9006 


8604 


8648 


86 9 3 


8 7 3 7 


8782 


8826 


8871 


8916 


8960 


45 


97 5 


9049 


9094 


9 i38 


9i83 9227 


9272 


9 3i6 


936i 


94o5 


45 


976 


9430 


9494 9339 
9939; 99S3 


9 583 


9628, 9672 


97H 


9761 


9806 


.9830 


44 


91 l 
978 


9893 


*° 2 S 


eo 7 2 » 1I7 


•161 


°2o6 


«25o 


•294 


44 


990339 


o3b3 


0428 


0472 


o5i6j o56i 


o6o5j o65o 


0694 


0738 


44 


979 


07S3 


0827 


0S71 


0916 


0960 1004 


1049' 1093 


1 1 37 


1182 


44 


980 


991226 


1270 


i3i5 


1359 


i4o3 1448 


1492 


i536 i58o 


i6 2 5 


44 


981 


1669 


I 7 i3 


1758 


^802 


1846 1890 


1935 


1979 


2023 


2067 


44 


982 


21 1 1 


2i56 


2200 


2244 


2288 2333 


2377 


2421 


2465 


25o9 


44 


983 


2J34 


2698 


2642 


2686 


2730 2774 


2819 


2863 


2907 


295i 
33g2 


44 


984 


2 99 5 


3o39 


3o83 


3127 


3172: 32i6 


326o 


33o4 


3348 


44 


985 


3436 


3480 


3524 


3568 


36i3 3657 


3701 


3 7 45 


3789 


3833 


44 


986' 


38 7 7 


3921 
436 1 


3965 


4009 


4o53j 4097 


4i4i 


4i85 


4229 


4273 


44 


987 


43i7 


44o5 


4449 


4493 


4537 


458i 


4625 


4669 


47i3 


44 


988 


4757 

5ig6 

995635 


4801 


4845 


4889 


4933 
53 7 2 


4977 


5021 


5o65 


5io8 


5i52 


44 


989 


6240 


5284 


5328 


5416 


5460 


55o4 


5547 


5591 


44 


99<r 


56 79 


5723 


5767 


58ii 


5854 


58 9 8 


5942 


5 9 86 


6o3o 


44 


99 1 


6074 


6,17 


6161 


62o5 


6249 


6293 


6337 

6774 


638o 


6424 


6468 


'44 


992 


65i2 


6555 


6599 


6643 


6687 


6 7 3i 


6818 


6862 


| 6906 


44 


99 3 


6949 


6993 7 o3 7 

743o 7474 


7080 


7124 


7168 


7212 


7255 


7299 

77 36 


1 7343 


44 


994 


7?6 


7517 


756i 


7605 


7648 


7692 


1 7779 


44 


993 


7823 


7867! 7910 


7934 


7998 
8434 


8041 


8o85 


8129 


8172 


1 8216 


44 


996 


8259 
8696 
9i3i 


83o3, 8347 


83 9 o 


8477 


852i 


8564 


8608 


j 8652 


44 


997 


8739 1 8782 


8826 


8869 8 9 i3 
93o5 9348 


8o56 

9 3 9 2 


9000 


9043 


9087 


44 


998 


9H4 9 21 . 8 


9261 


9435 


9479 
99 i3 


1 9 5 22 


44 


999 

N. 


9565 


9609 9652 


9696 


9739I 9783 


9826 


9870 


j 9957 


1 43 





I 


2 


3 


4 | 5 


6 


7 


8 


1 9' l"JX 



A TABLE 



OF 



LOGARITHMIC 
SINES AND TANGENTS 



FOR EVERY 



DEGREE AND MINUTE 
OF THE QUADRANT. 



Remark. The minutes in the left-hand column of 
each page, increasing downwards, belong to the de- 
grees at the top ; and those increasing upwards, in the 
right-hand column, belong to the degrees below. 



17 



18 


(0 


DEGREES-) A TABLE 


OF LOGARITHMIC 




M. 




Sine 

0-000000 


D. 


Cosine ! D. 


Tang. 


D. 


Cotang. 






io- 000000; 


o- 000000 




Infinite. 


60 


I 


6 -4637 20 


D017 * 17 


000000' 


00 


6-463726 


5017= 17 


1 3 536274 


% 


2 


764736 


2934 


85 


000000 


00 


764706 


2934 


83 


235244 


3 


940847 


2082 


3, 


000000 




00 


940847 


2082 


3i 


009 1 53 


57 


4 


7' 062786 


161D 


17 


000000 




00 


7-060786 


i6i5 


17 


12-034214 


56 


5 


162696 


i3i 9 


68 


000000 




00 


162696 


'1319 


60 


837304 


55 


6 


241877 


Hi5 


V 


9 . 999999 




01 


241878 


I I ID 


70 


758122 


54 


I 


308824 


966 


53 


999999 




01 


3o8820 


It 


5,5 


691 175I 53 
633i83i 52 


3668 1 6 


852 


54 


999999 




01 


366817 


54 


9 


417968 


762 


63 


999999 




01 


41797° 


762 


63 


582o3oi 5i 


10 


463725 


689 


ss 


999998 




01 


463727 


689 


88 


536273I 5o 


u 


7«5o5i 18 


629 


81 


9-999998 




01 


7«5o5i2o 


629 


8i 


1 2 -494880 49 
457091 1 48 


12 


542906 


5 79 


36 


999997 




CI 


542909 


5 79 


33 


i3 


577668 


536 


4i 


999997 




01 


677672 


536 


42 


422328! 47 


.'4 


609853 


499 


38 


999996 




01 


609857 


499 


3c 


390 1 431 46 


i5 


63 9 8i6 


467 
438 


14 


999996 




01 


639820 


467 


10 


36oi8o| 45 


16 


667845 


81 


999995 




01 


667849 


438 


82 


332 1 5t j 44 


l l 


694173 


4i3 


72 


999995 




01 


694179 


4i3 


73 


3o582ij 43 


18 


718997 


391 


35 


999994 




01 


7 1 9004 


3 9 i 


36 


280997 I 42 
257516 41 


*9 


742477 


3 7 i 


27 


999993 




01 


. 742484 


3 7 i 


28 


20 


764754 


353 


i5 


999993 




01 


764761 


35i 


36 


235239] 4° 


21 


7-785943 


336 


72 


9-999992 




01 


7=780951 


336 


73 


12«2i4o49i 39 
i 9 3845j 38 


22 


806146 


321 


7 5 


999991 




01 


8061 55 


321 


76 


23 


82545i 


3o8 


o5 


999990 
999989 




01 


825460 


3o8 


06 


1745401 37 
i56o56j 36 


24 


843934 


2 9 5 


47 




02 


843944 


295 


49 


2D 


861662 


283 


88 


999988 




02 


861674 


283 


90 
18 


138326! 35 


26 


878695 


2 7 3- 


H 


999988 




02 


878708 


2 7 3 


121292J 34 


27 


890085 


263 


23 


999987 




02 


895099 


263 


25 


1 0490 1 i 33 


28 


910879 


253 


99 


999986 




02 


9 1 0894 


254 


01 


089106 32 


29 


9261 19 


245 


38 


999985 




02 


926134 


245 


40 


073866 3i 
059142 3o 


3o 


940842 


2 3 7 


33 


999983 




02 


94o858 


23 7 


35 


3i 


7-955082 


229 


80 


9-999982 




02 


7-955ioo 


229 


81 


1 2 • 044900 29 


32 


968870 


222 


73 


999981 




02 


968889 


222 


75 


o3 1 1 1 1 


28 


33 


982233 


216 


08 


999980 




02 


982253 


216 


10 


017747 


ll 


34 


9 9 5io8 
8-007787 


209 


81 


999979 




02 


995219 


209 

203 


83 


0P4781 


35 


203 


90 


999977 




02 


8-007809 
020046 


92 


11-992191 


25 


36 


020021 


198 


3i 


999976 




02 


198 


33 


979955 


24 


37 


o3ioi9 
0435OI 


1 93 


02 


999975 
999973 




02 


o3ig45 


193 

188 


o5 


9 68o55 


23 


38 


188 


01 




02 


043527 


o3 


956473) 22 


39 


054781 


1 83 


25 


999972 




02 


054809 


1 83 


37 


945191! zz 


40 


065776 


178 


72 


999971 




02 


o658o6 


178 


^4 


934IQ4J 20 


41 


8-076500 


174 


41 


9-999969 




02 


8-07653i 


174 


44 


11-923469] 10 
9i3oo3| 18 


42 


086965 


170 


3i 


999968' 


02 


086997 


170 


34 


43 


097183 


• 166 


3 9 


999Q66; 


02 


097217 


166 


12 


O02783| 17 


44 


107167 


162 


65 


999964! 


03 


107202 


162 


68 


892707 16 

883o3 7 i5 


45 


1 16926 


159 


08 


099963 


o3 


1 1 6963 


i5^ 


!0 


46 


126471 


i55 


66 


Q99961 




o3 


i265io 


i55 


68 


873490! 14 


3 


i358io 


152 


38 


999v 5 9 




o3 


i3585i 


152 


41 


864149I 1 3 


144953 


149 


24 


999958 




o3 


i44qq6 
153.902 


149 


27 


855oo4l 12 


49 


153907 


146 


22 


9Q9956 




o3 


146 


27 


846048 


II 


5o 


162681 


143 


33 


999954; 


o3 


162727 


143 36 


837273 


iO 


5i 


8-171280 


140 


54 


9 -99995 2 ( 


o3 


8-171328 


140 57 


11-828672 


I 


52 


179713 


i3 7 


86 


999950 


o3 


179763 


137-90 


820237 


53 


187985 


i35 


29 


999948; 


o3 


i88o36 


135-32 


81 1964 
8o3844 


I 


54 


196102 


i3a 


80 


999946 


o3 


196156 


132-84 


55 


204070 


i3o 


41 


999944; 


o3 


204126 


i3o-44 


795874 


5 


56 


21 1895 


128 


10 


0Q9942 


04 


21 1953 


128-14 


788047 


4 


57 


219081 


125 


87 


999940 


04 


219641 


125-90 


78o35o 

772800 


3 


58 


227134 


123 


72 


999938, 


04 


227195 


123-76 


2 


5 9 


234557 


121 


64 


999936 


04 


234621 


121-68 


765379 


1 


60 


24i855 


119 


63 


999934: 


04 


241921 


119-67 


758079 








Cosine 


D. 


Sine |89° 


Cotang. 


D. 


Tang. I M. 





SINES 


AND TANGENTS 


^1 DEGREE.) 




19 


M. 


Sine 


D. , 


Cosine 


D. 


Tang. 


D. 


Cotang. 







8-24i855 


119-63 


9.999934 


.04 


8-241921 


119 


67 


11-758079 


60 


i 


249033 


117 


68 


999932 


• 04 


249102 


117 


72 


750898 


u 


2 


206094 


n5 


80 


999929 


.04 


256i65 


n5 


84 


743835 


3 


263o42 


n3 


98 


999927 


.04 


263u5 


114 


02 


736885 


57 


4 


269881 


112 


21 


999920 


• 04 


269956 


112 


25 


730044 


56 


5 


2">66i4 


no 


5o 


99992 2 


.04 


276691 


no 


54 


723309 


55 


6 


283243 


108 


83 


999920 


•04 


283323 


108 


87 


716677 


54 


7 


289773 


107 


21 


999918 


• 04 


289856 


107 


26 


710144 


53 


8 


296207 


100 


65 


999915 


■ 04 


296292 


jo5 


70 


703708 


52 


9 


302046 


104 


i3 


9999 l3 


•04 


3026J4 


104 


18 


697366 


5i 


10 


308794 


102 


66 


999910 


.04 


308884 


102 


70 


691 1 16 


5o 


ii 


8-3i4904 


101 


22 


9-999907 


•04 


8 • 3 1 5o46 


101 


26 


11.684954 


49 


12 


321027 


2 


82 


999905 


•04 


32II22 


99 


87 


678878 


48 


i3 


327016 


47 


999902 


.04 


327114 


98 


5i 


672886 


47 


14 


332924 


97 


14 


999899 


-o5 


333o25 


97 


19 


666975 


46 


i5 


338703 


90 


86 


999897 


• o5 


338856 


9 5 


90 


661 144 


45 


16 


344004 


94 


60 


999894 


• 00 


344610 


94 


65 


655390 


44 


12 


3ooi8i 


93 


38 


999891 


-o5 


300289 


93 


43 


6497 1 1 


43 


355783 


92 


o3 


999888 


• 00 


355893 


92 


24 


644io5 


42 


"9 


36i3i5 


9 1 


999885 


• o5 


36i43o 


91 


08 


638570 


4i 


20 


366777 


89 


90 


999882 


•o5 


3668 9 5 


89 


9 5 


633 1 o5 


40 


21 


8-372171 


88 


80 


9.999879 


• o5 


8-372292 


88 


85 


11-627708 


39 


22 


377499 


87 


72 


999876 


-o5 


377622 


87 


77 


622378 


38 


23 


382762 


86 


67 


999873 


• o5 


382889 


86 


72 


617m 


37 


24 


387962 


85 


64 


999870 


• 00 


388o 9 2 
393234 


85 


70 


611908 


36 


25 


393101 


84 


64 


999867 


• 00 


84 


70 


606766 


35 


26 


398179 


83 


66 


999864 


• 00 


3 9 83i5 


83 


7 1 


6oi685 


34 


:2 


403199 


82 


7 1 


999861 


-o5 


4o3338 


82 


76 


596662 


33 


408161 


81 


77 


999808 


• o5 


4o83o4 


81 


82 


591696 

586787 


32 


29 


4i3o68 


80 


86 


999854 


• o5 


4i32i3 


80 


9i 


3i 


3o 


417919 


79 


96 


999851 


• 06 


418068 


80 


02 


58i 9 32 


3o 


3i 


8-422717 


79 


09 


9.999848 


.06 


8.422869 


]l 


14 


n-577i3i 


29 


32 


427462 


78 


23 


999844 


.06 


427618 


3o 


572382 


23 


33 


432 1 56 


77 


40 


999841 


.06 


4323i5 


77 


45 


56 7 685 


27 


34 


4368oo 


76 


57 


999838 


• 06 


436962 


76 


63 


563o38 


26 


35 


44i394 


75 


77 


999834 


.06 


44i56o 


75 


83 


558440 


25 


36 


445941 


74 


99 


999831 


• 06 


446 1 1 


75 


o5 


553890 


24 


3? 


45o44o 


74 


22 


999827 


• 06 


4006 1 3 


74 


28 


549387 


23 


38 


454893 


73 


46 


999823 


• 06 


455070 


73 


52 


544q3o 


22 


3 9 


409301 
463665 


72 


73 


999820 


• 06 


459481 


72 


79 


54o5 1 9 


21 


4o 


72 


00 


9998161 


• 06 


463849 


72 


06 


536i5i 


20 


4i 


8-467985 


7i 


29 


9.999812 


.06 


8.468172 


7i 


35 


ii-53i828 


S 


42 


472263 


70 


60 


999809 
999803 


.06 


472454 


70 


66 


527546 


43 


476498 


69 


9i 


.06 


476693 


69 


f, 


5233o7 


17 


44 


480693 


69 


24 


999801 


.06 


480892 


it 


5 1 91 08 


16 


45 


484848 


68 


5 9 


999797 


.07 


485o5o 


65 


514950 
5io83o 


i5 


46 


488 9 63 


67 


94 


999793 


.07 


489170 


68 


01 


14 


47 


493040 


67 


3i 


999790 


.07 


49325o 


67 


38 


506750 


i3 


48 


497078 


66 


69 


999786 


.07 


/ 9 72 9 3 


66 


76 


502707 


12 


49 


5oio8o 


66 


08 


999782 


•07 


501298 


66 


i5 


498702 


11 


5o 


5o5o45 


65 


48 


999778 


•07 


5o5267 


65 


55 


494733 


10 


5i 


8- 5o8 97 4 


64 


89 


9*999774 


.07 


8-509200 


64 


96 


11-490800 


i 


52 


512867 


64 


3i 


999769 


•07 


513098 


64 


3 9 


486902 


53 


516726 


63 


75 


999760 


•07 


5 1 696 1 


63 


82 


483o39 


7 


54 


52o55i 


63 


19 


999761 


• 07 


520790 


63 


26 


479210 
475414 


6 


55 


524343 


62 


64 


999707 


•07 


524586 


62 


72 


5 


56 


528102 


62 


11 


999703 


•07 


528349 


62 


18 


47i65i 


4 


n 


53i828 


61 


58 


999748 


.07 


532o8o 


61 


65 


467920 


3 


535523 


61 


06 


999744 


•07 


535779 


61 


i3 


464221 1 2 


5 9 


539186 


60 


55 


999740 


• 07 


539447 


60 


62 


46o553| 1 


60 


542819 


60 


04 


999735 


•07 
88° 


543o84 


60 


12 


4569 1 6| 


j | Cosine 


D 


. I Sine 


Cotang. 


D. 


Tang 1 M. 



20 


(2 


DEGREES.) A TABLE OF LOGARITHMIC 




M. 


. Sine | 


D - I 


Cosine 


D. 


Tang. ( 


D. 


Cotang. | 




o 


8.542819 i 


60-04 1 


9-999735 


• 07 


8-543o84 


60-12 


1 1 -4569161 


60 


i 


546422 


59-55 I 


999731 


.07 


546691 


59-62 


453309) 5o 

449732 58 


2 


549993 
553539 


5 9 -o6 


999726 


.07 


55o268 


59.14 


: 3 


58-58 


999722 


• 08 


5538i 7 


58-66 


446i83! 57 


1 4 


557054 


58-n 


9997 ' 7 


.08 


557336 


58-i 9 


442664I 56 


5 


56o54o 


57-65 


9997 1 3 


.08 


560828 


5 7 . 7 3 


439172 55 


6 


563999 


5 7 -i 9 


999708 


•08 


564291 


57.27 


433709! 54 


I 


56743i 


56-74 


999704! 


• 08 


567727 


56-82 


4322731 53 


570836 


56-3o 


999699 


• 08 


57 1 1 37 


56-38 


428863 52 


9 


574214 


55-87 


999694 


.08 


574520 


55- Q 5 


425480I 5i 


10 


577566 


55-44 


999689 


.08 


577877 
8-58i2o8 


55-52 


422I23| 5o 


ii 


8- 58o8 9 2 


55-02 


9-999683 


.08 


55- 10 


11-418792! 49 
41 5486 1 48 


12 


584ig3 


54-6o 


999680 


.08 


5845.4 


54-68 


i3 


587469 


54-19 


999675 


.08 


587795 


54-27 


4i22o5| 47 


14 


590721 


53-79 


999670 


• 08 


59 1 00 1 


53-87 


4089491 46 


i5 


593948 


53-3 9 


999660 


• 08 


5 9 4283 


53-47 


405717 45 


16 


597152 


53-oo 


999660 


•08 


597492 


53- 08 


4o25o8' 44 


\l 


6oo332 


52-6i 


999655 


• 08 


6006T' 


52-70 


399323) 43 


603489 


52-23 


99yo5o 


.08 


6o383o 


52-32 


396161] 42 


*9 


60662J 


5i-86 


999645 


•09 


606978 


5i-o4 


393022I 4i 


20 


609734 


5i-49 


999640 
9.999635 


-09 


610094 


5i-58 


3899061 4o • 


21 


8-612823 


5l-I2 


• 09 


8.613189 


5l-2I 


1 1 -3868i 1 j 3g 

383738 38 


22 


61D891 


50-76 


999629 


.09 


616262 


5o-85 


23 


618937 


5o-4i 


999624, 


.09 


6i93i3 


5o-5o 


380687 3 7 


24 


621962 


5o-o6 


999619 


.09 


622343 


5o-i5 


3 77 65 7 36 


25 


624963 


49-72 


999614 


.09 


625352 


49-81 


374648 35 


26 


627948 


49-38 


999608 


.09 


628340 


49-47 


371660J 34 


27 


630911 


4q-o4 
48-71 


999603 


.09 


63i3o8 


49- 13 


368692I 33 


28 


633854 


999597 


.09 


634256 


48-8o 


365744 32 


29. 


636776 


48- 3 9 


999592 


.09 


637184 


48-48 


362816 3i 


3o 


639680 


48- 06 


999586 


.09 


640093 


48-16 


3599071 3o 


3i 


8-642563 


47-75 


9.999581 


.09 


8.642982 


47-84 


11 .357018 


29 


32 


64542S 


47-43 


999575 


.09 


645853 


47-53 


354U7 


28 


33 


648274 


47-12 


999570 


.09 


648704 
65i537 


47-22 


351296 


27 


34 


65uo2 


46-82 


999564 


.09 


46-91 


348463 


26 


35 


653911 


46-52 


999558 


•10 


654352 


46-6i 


345648 


25 


36 


606702 


46-22 


999553 


•10 


657149 


46- 3 1 


34285i 


24 


37 


659475 


45.92 


999547 


• 10 


659928 


46-02 


340072 


23 


38 


662230 


45-63 


999541 


• 10 


662689 


' 45- 7 3 


3373 1 1 


22 


3 9 


664968 


45-35 


999535 


• 10 


665433 


45-44 


334567 


21 


4o 


667689 


45- 06 


999529 


.10 


668160 


45-26 


33i84o 


20 


4i 


8-670393 


44-79 


9«99g524 


• 10 


8-670870 


44-88 


ii'32gi3o 


19 


42 


673080 


44-5i 


999518 


•10 


673563 


44-6i 


326437 


18 


43 


675751 


44-24 


9995 1 2 


•10 


676239 


44-34 


323761 


'7 


44 


678405 


43-97 


999506 


•10 


678900 


44-17 


321100 


16, 


45 


681043 


43-70 


999500 


•10 


68i544 


43- 80 


3 18456 


i5 


46 


683665 


43-44 


999493 
999487 


•10 


684172 


43-54 


3i5828 


14 


47 


686272 


43- 18 


• 10 


686784 


43-28 


3i32i6 


i3 


48 


688863 


42-92 


99948i 


•10 


68 9 38i 


43- o3 


3 1 06 1 9 


12 


49 


691438 


42-67 


999475 


•10 


691963 


42-77 


3o8o37 


11 


5o 


693998 


42-42 


999469 


•10 


694029 


42-52 


3o547i 


10 


5i 


8 • 696043 


42-17 


9.999463 


•II 


8-697081 


42-28 


11-302919 


8 


52 


699073 


41-92 


999456 


• II 


699617 


42 -o3 


3oo383 


53 


701589 


41-68 


99945o 


• I I 


702139 


41-79 


297861 


7 


54 


704090 


41-44 


999443 


• II 


704646 


4i-55 


295354 


6 


55 


7o65 77 


4I-2I 


999437 


. 11 


707140 


4i-32 


292860 


5 


56 


709049 


40-97 


999431 


• II 


709618 


41-08 


2oo382 


4 


57 


711507 


40-74 


999424 


• 1 1 


712083 


40-85 


287917 


3 


58 


713902 


40 -5i 


999418 


•I I 


714534 


40-62 


285465 


2 


5 9 


7 i6383 


40-29 


9994 n 


•II 


716972 


40-40 


283028 


1 


60 


718800 


4o 06 


999404 

Sine 


•II 


719396 


40-17 


280604 







Cosine 


87° 


Cotang. 


D. 


Tang. 


M. 





SINES AND TANGENTS 


. (3 DEGREES.' 




21 


M. 


1 Sine 


D. 


Cosine 


D. 


Tansr. 


D. 


CV>tang.« 





8-718800 


4o- 06 


9-999404 




8.719396 


40-17 


11-280604 60 


i 


721204 


3 9 -84 


999398 




721806 


39.95 


278194 5o 
275796; 58 


2 


7235q5 


3 9 -62 


999391 




724204 


39-74 


3 


723972 


39-41 


999384 




726088 


39.52 


273412 


57 
56 


4 


728337 


39-19 


999378 




728959 


39-3o 


27 1 04 1 


5 


73o688 


3S- 9 8 


999371 




73i3n 


39-09 


268683 


55 


6 


i 733027 


38-77 


999364 


•12 


733663 


38- 89 


266337 


54 


I 


1 735354 


38-5 7 


999357 


•12 


735996 


38-68 


264004 


53 


737667 


38-36 


999350 


•12 


7383 1 7 


38-48 


26i683 


52 


9 


739969 


38- 16 


999 3 43 


•12 


740626 


38-27 


259374 


5i 


10 


742209 


37-96 


999336 


• 12 


742922 


38-o 7 


257078 5o 


ii 


8-744536 


37-76 


9-999329 


•12 


8-745207 


37-87 


11 -254793 i 49 

25232I| 48 


12 


746802 


37-56 


999322 


•12 


747479 


37-68 


i3 


749055 


37-37 


9993 1 5 


•12 


749740 


37-49 


2502601 47 


14 


751297 


37-17 


999308 


•12 


731989 


37-29 


24801 1 1 46 


i5 


7535-28 


36 -98 


999301 


•12 


754227 


37- 10 


a45773 i 45 


16 


755747 


36-79 


999294 


•12 


756453 


36- 9 a 


243547; 44 


3 


757955 


36- 61 


999286 


•12 


758668 


36-73 


24i332, 43 


760 1 5 1 


36-42 


999279 


•12 


760872 


36-55 


239128 42 


19 


762337 


36-24 


999272 


•12 


763o65 


36-36 


236935; 41 


20 


7645 1 1 


36- 06 


999260 


•12 


765246 


36-i3 


234754 1 4o 


21 


8-766675 


35-88 


9-999257 


•12 


8.767417 


36-oo 


r 1 -232583 39 


22 


768828 


35.70 


999250 


•13 


769578 


35-83 


23o422| 3& 


23 


770970 


35-53 


999242 


• i3 


771727 


35-65 


228273, 37 


24 


773ioi 


35-35 


999235 


• i3 


773866 


35-48 


226134' 36 


25 


775223 


35.i8 


999227 


•i3 


775995 


35-3i 


2240055 35 


26 


777333 


35-01 


999220 


• i3 


773II4 


35-14 


221886; 34 


27 


779^34 


34-84 


999212 


-i3 


780222 


34-97 
34- 80 


219778 33 
217680! 32 


28 


781524 


34-67 


999203 


• i3 


782320 


29 


7836o5 


34-5i 


999107 
999189 


• i3 


784408 


34-64 


215592; 3 1 


3o 


o 7 856 7 5 


34-3i 


• i3 


786486 


34-47 


2i35i4 ! 3o 


3i 


8-787736 


34-iS 


9.999181 


• i3 


8-788554 


34-3i 


11-211446 29 
209387 j 28 


32 


789787 


34-02 


999174 


• i3 


790613 


34- 15 


33 


79182S 


33-S6 


999166 


• i3 


792662 


33.99 


207338; 27 


34 


793Si 9 


33-70 


999158 


..3 


794701 


33-83 


205299 1 26 


35 


79 588 1 


33-54 


999 1 5o 


• i3 


796731 


33-63 


2o326o 25 


36 


797'V 


33.39 


999*42 


• i3 


79S752 


33-52 


201248! 24 


37 


799 8 97 

801892 


33-23 


999i34 


• i3 


800763 


33-3 7 


199237 23 


38 


33.o8 


999126 


• i3 


802765 


33-22 


197235 22 


3 9 


8o3S 7 6 


32-93 


9991 18 


•i3 


8o4758 


33-07 


195242 21 


40 


8o5S5 2 


32-78 


9991 10 


•i3 


806742 


32-92 


193238; 20 


4i 


8-807819 


32-63 


9.999102 


-i3 


8-808717 


32- 7 8 


11.191283! 19 
189317 18 


42 


809777 


32-49 


999094 


•14 


8io683 


32-62 


43 


811726 


32-34 


999086 


- l A 


812641 


32-48 


I87359 1 17 


44 


8 1 366 7 


32-19 


999077 


•14 


8i458 9 


32-33 


1 854 1 1 


16 


45 


815599 


32-o5 


999069 


•14 


8i652 9 


32-19 


1 8347 1 


i5 


46 


817522 


3i -91 


999061 


• 14 


818461 


32-03 


i8i53g 


i4 


2 


8i 9 436 


3i-77 


999053 


• 14 


820384 


3i -91 


179616' i3 


82i343 


3i-63 


999044 


• 14 


822298 


31.77 


177702 


12 


49 


823240 


3i-49 


999036 


• 14 


824205 


3i-63 


175795 


11 


5o 


825i3o 


3i-35 


999027 


• 14 


826103 


3i-5o 


173897 


10 


5i 


8-827011 


3l-22 


9.999019 


• 14 


8-827992 


3i-36 


11.172008 





52 


828884 


3i-o8 


999010 


• 14 


829874 


3i-23 


170126 


8 


53 


830749 


3o-95 


999002 


•14 


83 1 748 


3i- 10 


168252 


-1 


54 


832607 


3o-82 


998993 


• 14 


8336i3 


3o-o6 


166387 


6 


55 


834456 


3o-6g 


998984 


•14 


835471 


3o-83 


164539 


5 


56 


836297 


3o-56 


998976 


•14 


837321 


30-70 


162679 


4 


5 7 


838i3o 


3o-43 


99S967 


• i5 


83 9 i63 


3o-57 


160S37 


3 


58 


839956 


3o-3o 


99 8 9 58 


• i5 


840998 


3o-45 


159002 


2 


5 9 


841774 
843585 


3o- 17 


9q8o5o 


•i5 


842825 


3o-32 


157175 


1 


60 


3o-oo 


998941 


• i5 


844644 


3o-i9 


155356 


1 




Cosine 


r> 


Sine 


SG° 


Cotang. 


D. 


Tans. !M. j 



22 


(1 


DEGREES.) A TABLE 


OF LJGARITFiMIC 




m. i 


Sine [ 


D. 


Cosine | 


D. 1 Tang. 


D. 


Cotang. 


! 


o ! 


3- 843585 


3o-o5 


9-908941 


•i5 8-844644 


3o- 19 


ii-i55356 


~ST 


I 


845387 


29-02 
: 9 -8o 


998932 | 


•i5 846455 


30-07 


153545 


59 


a 


847183 


998923 


•i5 84S260 


29-95 


151740 


58 


3 


848971 


29-67 


998914 


•i5| 85oo57 


29-82 


149943 


5 7 


4 


85o 7 5i 


29-55 


998905 
998896 
998887 


•i5 ! 85i846 


29.70 


U8i54 


56 


5 


852525 


29-43 


•i5 853628 


29.58 


146372 


55 


6 


854291 


29-31 


•i5 8554o3 


29.46 


144597 


54 I 


7 


856049 


29-19 


998878 


• i5 


857171 


.29-35 


142829 
141068 


53 | 


8 


807801 


20-07 


998869 


• i5 


858 9 32 


29-23 


52 1 


9 


85g546 


28-96 


998860 


• i5 


860686 


29-11 


i3g3i4 


5r 


10 


861283 


28-84 


998851 


• i5 


862433 


29-00 


137567 


5o 


ii 


8-863oi4 


28-73 


9-998841 


• i5 8-864173 


28-88 


11-135827 


4 2 


12 


864738 


28-61 


998832 


•i5 865 9 o6 


28-77 


134094 


48 


i3 


866455 


28 -5o 


998823 


.16 867632 


28-66 


132368 


47 


.4 


8681 65 


28-39 
28-28 


998813 


.16 86 9 35i 


28-54 


1 30649 


46 


i5 


869868 


998804 


.16 871064 


28.43 


I28 9 36 


45 


16 


87 1 565 


28-17 


998795 


•16 872770 


28-32 


127230 


44 


'7 


8 7 3255 


28-06 


998785 


.16 874469 


28-21 


i2553i 


43 


18 


874938 


27.95 


998776 


.16 876162 


28-11 


123838 


42 


19 


876615 


27-86 


998766 


.16 


877849 


28-00 


I22l5l 


4i 


20 


878285 


27-73 


998757 


.16 


879529 


27-89 


I 20471 


4o 


21 


8-879949 


27-63 


9-998747 


.16 


8-881202 


27.79 


II I 1 8798 


39 


22 


881607 


27-52 


998738 


.16 


882869 


27.68 


H7i3i 


38 


23 


883258 


27-42 


998728 


.16 


88453o 


27.58 


1 1 5470 


37 


24 


884903 


27 -3i 


998718 


•16 


886i85 


27-47 


n38i5 


36 


2D 


886542 


27-21 


998708 


•16 


887833 


27.37 


112167 


35 


26 


888174 


27-11 


998699 


.16 


889476 


27-27 


no524 


34 


27 


889801 


27-00 


998689 


•'16 


891112 


27-17 


108888 


33 


28 


891421 


26-90 


998679 


• 16 


892742 


27.07 


107258 


32 


29 


8 9 3o35 


26-80 


998669 


•17 


8 9 4366 


26.97 


io5634 


3i 


3o 


894643 


26-70 


998659 


•n 


8 9 5 9 84 
8 -897696 


26.87 


104016 


3o 


3i 


8-896246 


26-60 


9-998649 


•17 


26.77 


11 -102404! 29 


32 


807842 


26-5i 


998639 


•n 


899203 


26-67 


100797 


28 


33 


899432 


26-41 


998629 


•17 


900803 


26-58 


099197 


27 


34 


901017 


26-3i 


998619 


•n 


902398 


26-48 


097602 


26 


35 


902596 


26-22 


998609 


•H 


903987 
905570 


26-38 


096013 


25 


36 


904169 


26-12 


998599 


• '7 


26-29 


094430 


24 


37 


905736 


26 -o3 


99 858 9 


•n 


907147 


26-20 


092853 


23 


38 


907297 


25 - 9 3 


998578 


•17 


9087*9 
910285 


26-10 


091281 


22 


3 9 


9 o8853 


25-84 


998568 


•n 


26-01 


o8 97 i5 


21 


4o 


9 1 0404 


25- 7 5 


99 8558 


•n 


91 1846 


25-92 


o88i54 


20 


4i 


8-91 1949 
913488 


25-66 


9-998548 


•n 


8-913401 


25-83 


11 -086599 


\l 


42 


25-56 


99 853 7 


.17 


9U951 


25.74 


o85o4g 
o835o5 


43 


9i5o22 


25-47 


998527 


:3 


916495 


25-65 


\l 


44 


9i655o 


25-38 


99 85 1 6 


9180J4 


25-56 


081966 


45 


918073 


25-29 


998506 


.18 


919568 


25-47 


080432 


i5 


46 


919591 


25-20 


99 8 495 
998485 


.18 


921096 


25-38 


078904 

077381 


14 


47 


921 1 o3 


25-12 


.18 


922619 


25-3o 


i3 


48 


922610 


25-o3 


998474 


.18 


924136 


25-21 


075864 


12 


49 


924112 


24-Q4 


998464 


.18 


925649 


25-12 


07435i 


11 


5o 


925609 


24-86 


998453 


•18 927156 


25 -o3 


072844 


10 


5i 


8-927100 


24-77 


9-998442 


•181 8-928658 


24-9 5 


1 i- 07 1 342 


9 


52 


9 2858 7 


24-69 


998431 


.18 


93oi55 


24-86 


069845 8 


53 


930068 


24-60 


998421 


• 18 


931647 


24-78 


068353 


7 


54 


93 1 544 


24-52 


998410 


.18 


933i34 


24-70 


066866 


6 


55 


933oi5 


24-43 


998399 
99 8388 


•18 


934616 


24-61 


o65384 


5 


56 


934481 


24-35 


.18 


936093 


24-53 


063907 


4 


57 


935942 


24-27 


998377 


• 18 


937565 


24-45 


062435 


58 


937398 


24-19 


998366 


.18 


93go32 


24-37 


06096S 
o5g5o6 
o58o48 


2 


5 9 


93885o 


2411 


998355 


• 18 


940494 


24 -3o 


1 


60 


940296 


?4-o3 


998344 


.18 


941952 


24-21 





! 


Cosine 


D. 


Sine 


85°i Cotansr. 


1 D. 


Tansr; M. 





SINES AND TANGENTS. 


(5 DEGREES.) 


2 


M. 


Bine 


D. 


Cosine i I). | 


Tang. 1 


D. 


Cotang 1 . I 





8-940296 
94H38 


24 -o3 


9-998344 -19! 


8-9419321 


24-21 


1 1 • o58o48 60 


1 


23-94 
23 .87 


998333 




i9 : 


943404 1 


24-i3 


056596 59 
o55i48 58 


2 


943174 


,998322 




•9 


944852 


24-03 


3 


944606 


23-79 


99831 1 




19 


946295- 


23-C7 


o537o5 37 


4 


946034 


23-71 


998300 




19 


947734 


23-90 


o52266 56 


5 


947456 


23-63 


998289 




i9i 


949168 


i3-82 


o5o832 55 


6 


948874 


23 • 55 


998277 




19 


950597 


23-74 


049403 34 


7 


950287 


23-48 


998266 




19 


952021 


23-66 


04-979 53 


8 


951696 


23-4o 


998255 




«9 


953441 


23- 60 


046359 52 


9 


953ioo 


23-32 


998243 




19 


954856| 


23-5i 


o45 144! 5 1 


10 


954499 


23-23 


998232 




*9 


956267 


23-44 


043733 5o 


u 


8-955894 


23-17 


9.998220 
998209 




>9 


8-937674 


23;37 


11-042326- 49 


12 


957284 


23- 10 




19 


939075 


23-29 


040923! 48 


i3 


958670 


23-02 


998197 




*9- 


960473 


23-23 


0393271 47 


14 


960052 


22-93 


998186 




'9 


961866 


23- 14 


o38i34! 46 


i5 


961429 


22-88 


998174 




i9| 


963255 


23-07 


o36745i 45 


16 


962801 


22-80 


998163 




19! 


964639 


23-00 


o3536i; 44 


\l 


964 1 70 


22-73 


998i5i 




'9 


966019 


22-93 


o33 g 8i 43 


965534 


22-66 


998139 


201 


967394! 


22-86 


o3 26061 42 


*9 


966893 


22-59 


998128 


2o| 


9687661 


22-79 


o3i234' 41 


20 


968249 


22-32 


9981 16 




20] 


970133. 


22-71 


029867 40 


21 


8-969600 


22-44 


9-998104 




20, 


8-071496! 


22-65 


u-o285o4 39 


22 


970947 


22-38 


998092 




20| 


972835J 


22-57 


027145 38 


23 


972289 


22 -3l 


998080 


20 


974209I 


22 - 5l 


026791 37 


24 


973628 


22-24 


998068 
998056; 


20| 


97556o 


22-44 


024440 36 


20 


974962 


22-17 


20| 


976906 


22-37 


023094; 35 


26 


976293 


22-13 


998044 


20| 


978248 


22-30 


021752 34 


3 


977619 


22-o3 


99S032 




20; 


979 5 86 ; 


22-23 


020414. 33 


978941 


21-97 


998020 




20 


980921 


22- 17 


019079, 32 


o9 


980259 


21-90 
2i-83 


998008 




20 


982251 1 


22-10 


017749 3i 


3o 


981573 


997996 




20 


9835771 


22-04 


01642.3 io 


3i 


8-982883 


21-77 


9-997983! 


20 


8-984899 


21-97 


ii - c 1 5"i 1 1 29 


3 2 


984189 


21 -70 


997972 


20' 


9862171 


21 -91 


OI3783 28 


33 


983491 


21-63 


997939! 


20 


987532' 


21-84 


012468 27 


34 


986789 
98S083 


21-57 


9979^7, 


20 


9S8842 


21-78 


01 1 i58 26 


35 


21 -5o 


997935, 


21 


99c 149' 


21-71 


ooo85i 25 
008549 2 4 


36 


989374 


21-44 


997922| 


2i! 


99i45i! 


21 -65 


12 


990660 


21-38 


997910 


21 


992750I 


21-58 


007250 23 


99 '943 


2I-3l 


997897 




21 


994045 j 


21-52 


005935 22 


3 9 


993222 


21-25 


997885 




2I 1 


9953371 


21-46 


oo4663 2 1 


4o 


994497 


21-19 


997872 




21 j 


9966241 


21-40 


003376 20 


4i 


8-993768 


21-12 


9.997860 




21 


8-997908; 


21-34 


11-002092 19 


42 


997036 


21 -06 


99^847 




21 


999.881 


21-27 


000812 18 


43 


998299 


21-00 


997835 




21 


9-000465' 


21-21 


20-999535' 17 


44 


999560 


20-94 

20-8 7 


997822 




21. 


001738! 


21 -l5 


998262' 16 


45 


9-000816 


997809 




2i; 


003007J 


21-09 


996993 13 


46 


002069 


20-82 


997797 




2I i 


004272I 


21 -o3 


9957281 14 


47 


oo33i3 


20-76 


997784 




21 


oo5534| 


20- Qi 


994466 ' i3 


48 


oo4563 


20-70 


997771 




2I I 


006792, 


20-91 


993208' 12 


49 


oo58o5 


20-64 


997758 




2l! 


008047 


20-85 • 


991953' 11 


DO 


007044 


20-53 


997745 




21 


009298! 


20-80 


990702 10 


5i 


9.008278 


20-52 


9.997732 




2I l 


9-oio546| 


20-74 


10-989454 9 
988210 8 


52 


009510 


20-46 


997719 




21, 


01 1790 
oi3o3i 


20-68 


53 


010737 


20-40 


997706 




2-1 ! 


20-62 


986969 7 


54 


011962 


20-34 


997693 
997680 




22| 


014268I 


20-56 


9 85 7 32 6 


55 


oi3i82 


20-29 




22 


oi55o2, 


20-5l 


984498 5 


56 


014400 


20-23 


997667 




22' 


016732 


20-43 


983268 4 


11 


oi56i3 


20- 17 


997634 




22 


017959 


20 • 40 


982041 3 


016824 


20-12 


997641 




22 


019183, 


20-33 


980817 2 


5 9 


oi8o3i 


20- 06 


997628; 


22 


020403 


20-28 


979597 » 
978380 


6o 


019235 


20 • 00 


997614; 


22 


021620 


20-23 


lH 


Cosine 


D. " 


Sine '84°: 


Cotang. 1 


p. 


Tan-r nr 



29 



24 


(6 


DEGREES.) A TABLE 


OF LOGARITHMIC 


60 


M- 


Sine 


D. 


Cosine 


D. 


Tang. 


D. 


Cotang. 


o 


9-019235 


20 -00 


9-997614 


• 22 


9-021620 


20-23 


10-978380 


i 


020435 


J 9 


9 5 


997601 




22 


022834 


20 


'7 


977 166 1 5o 
975956 58 


2 


021632 


*9 


89 


997588 




22 


024044 


20 


11 


3 


022825 


l 9 


84 


997574 




22 


o2525i 


20 


06 


974749! *>1 


4 


024016 


! 9 


78 


997561 




22 


026455 


20 


00 


9735431 56 


5 


025203 


*9 


73 


997547 




2 2 


027655 


'9 


9 5 


972345 55 


6 


026386 


J 9 


67 


997534 




23 


028852 


19 


90 


971148! 54 


7 


027567 


J 9 


62 


997520 




23 


030046 


'9 


85 


969954 53 
968763 52 


8 


028744 


J 9 


57 


997507 




23 


031237 


*9 


79 


9 


029918 


'9 


5i 


997493 




23 


032425 


19 


74 


967575-! 5i 


10 


o3 1 089 


'9 


47 


997480 




23 


033609 


19 


69 


96639 1 | 5o 


i i 


9-032257 


'9 


4i 


9 • 997466 




23 


9-034791 


>9 


64 


10-965209! 4q 
96403 1 i 48 


12 


o3342i 


19 


36 


997452 




23 


035969 


'9 


58 


i3 


o34582 


'9 


3o 


997439 




23 


037144 


*9 


53 


9628561 47 


• 4 


035741 


] 9 


25 


997423 




23 


o383i6 


r 9 


48 


961684' 46 


i5 


o368 9 6 


'9 


20 


99741 1 




23 


039485 


J 9 


43 


9605 1 5 45 


16 


038048 


l 9 


i5 


997397] 


23 


0406 5 1 


J 9 


38 


95o349 1 44 
958 187 j 43 


\l 


039197 


19 


10 


997383 


23 


041813 


*9 


33 


040342 


I? 


o5 


997369 




23 


042973 


'9 


28 


9570271 42 


*9 


o4i485 


99 


99735a 




23 


0441 3o 


19 


23 


9558-joI 41 


20 


042625 


18 


94 


997341 




23 


045284 


*9 


18 


9547 16 j 40 


21 


9-043762 


18 


89 


9.997327 




24 


9-046434 


19 


i3 


IO-933566 1 39 
9 524i8| 38 


22 


044895 


18 


84 


9973 1 3 




24 


047582 


*9 


08 


23 


046026 


18 


79 


697299 
997283 




24 


048727 
049869 
o5iooo 


\l 


o3 


9 5i273] 37 


24 


o47i54 


18 


75 




24 


98 


95oi3i; 36 


2D 


048279 


18 


70 


997271 




24 


18 


ll 


948992 35 
947856 34 


26 


049400 


18 


-65 


997257 




24 


o52i44 


18 


11 


o5o5i9 


18 


•60 


997242 




24 


053277 


18 


84 


946723 33 


o5i635 


18 


•55 


997228 




24 


054407 


18 


79 


945593) 32 


29 


052749 


18 


5o 


997214 




24 


o55535 


18 


74 


944465 3 1 


3o 


o53859 


18 


45 


997199 




24 


o56659 


18 


70 


943341 3o 


3i 


9-054966 


18 


•4i 


9-997180 




24 


9-057781 


18 


65 


10-942219! 29 


32 


056071 


18 


•36 


997170 




24 


038900 


18 


53 


941 iooj 28 


33 


057172 
058271 


18 


3i 


997106 




24 


060016 


18 


939984! 27 


34 


18 


27 


997141 




24 


061 i3o 


18 


5i 


938870; 26 


35 


059367 


18 


22 


997 I2 7 




2-1 


062240 


18 


46 


937760' 25 


36 


060460 


18 


•n 


9971 12 




24 


o63348 


18 


42 


936652 


24 


37 


06 1 55 i 


18 


i3 


997°9 8 | 
997083 [ 


24 


o64453 


18 


37 


935547 


23 


38 


062639 


18 


08 


25 


o65556 


18 


36 


934444 


22 


3 9 


063724 


18 


04 


9970681 


25 


o66655 


1 8 


28 


933345 


21 


4o 


064806 


n 


•99 


997033 


2 3 


067752 


18 


24 


932248 


20 


4i 


9-o65885 


n 


•q4 


9 -997039 


25 


9.068846 


18 


! 9 


io-93ii54 


!? 


42 


066962 


H 


£ 


997024 




25 


069938 


18 


13 


930062 


43 


* o68o36 


17 


997009 




25 


071027 


18 


10 


928973 


n 


44 


069107 


n 


81 


996994 




2 J 


072113 


18 


06 


927887 


! 5 


45 


070176 


n 


77 


996979 




2 5 


073197 

074278 


18 


02 


926803 


i5 


46 


071242 


17 


72 


996964 




23 


17 


97 


925722 


14 


4i 


072306 


17 


63 


996949 




23 


075356 


n 


89 


924644 


i3 


48 


073366 


17 


63 


996934 




23 


076432 


17 


923568 


12 


49 


074424 


n 


5 9 


996919 




23 


077505 


'7 


84 


922495' 11 


DO 


075480 


17 


53 


996904 




23 


078576 


17 


80 


921424 10 


5i 


9-076533 


n 


5o 


9.996889 




25 


9-079644 


17 


76 


io-92o356 9 


52 


o 77 583 


H 


46 


996874 


25 


080710 


17 


72 


919290 8 


53 


078631 


17 


42 


9 9 6858j 


25 


081773 


17 


ol 


918227 7 


54 


079676 


'7 


38 


996843 


25 


oS-;833 


17 


9.1 7 167; 6 


55 


080719 


17 


33 


996828, 


25 


083891 


n 


5 9 


916109 5 


56 


081759 


'7 


29 


996812 


26 


084947 


17 


55 


9 1 5o53 ; 4 


^7 


082797 


'7 


25 


996797 


26 


086000 


'7 


5i 


914000 3 


58 


o83832 


17 


21 


996782 


26 


087050 


17 


47 


912950 2 


5 9 


084864 


17 


•7 


996766] 


26 


088098 


17 


43 


911002 1 


6o 


080894 


17 


i3 


996751 




26 


089144 


17-38 


9io856; 


L_. 


Cosine 


IX 


Sine 


8 


30 


Cotang. 


D. 


~?^zr 


M: 



BINES AND TANOENTS. (7 DEGREES.) 



26 



,<M. 


Sine 


D. 


Cosine 


D. 


Tang. 


D. 


Cotang. J 


o 


9-085894 


17- 13 


9 -99075 1 


•26 


9-089144 


17-38 


10-910856 


60 


I 


086922 


17-09 


996735 


.26 


09018' 


17-34 


909813 
908772 


5 9 


2 


087947 


17-04 


996720 


•26 


1 091228 


17 -3o 


58 


3 


088970 


17-00 


996704 


.26 


092266 


17-27 


907734 


57 


4 


089990 


16-96 


996688 


•26 


093302 


17-22 


906698 


5b 


5 


091008 


i6-q2 


99667J 


•26 


094336 


17.19 

17- 15 


905664 


55 


6 


092024 


16-88 


996657 


•26 


095367 


904633 


54 


I 


093037 


16-84 


996641 


■26 


096395 


1711 


9o36o5 


53 


094047 


16.80 


996625 


•26 


097422 


17-07 


902578 


52 


9 


09Do56 


16-76 


996610 


•26 


098446 


i7-o3 


901554 


5i 


10 


096062 


16-73 


996594 


•26 


099468 


1699 
16-95 


900532 


5o 


] i 


9*097065 


16.68 


9- 99 65 7 8 


•27 


9 • 1 00487 


10-899513 


% 


12 


098066 


16-65 


996562 


•27 


ioi5o4 


16-91 


898496 


i3 


099065 


16. 61 


996546 


•27 


102519 


16-87 


897481 


47 


H 


100062 


i6-5 7 


99653o 


•27 


io3532 


16-84 


8964681 46 


]5 


ioio56 


16-53 


9965 1 4 


•27 


104542 


16-80 


895458 45 


16 


102048 


16-49 


996498 


.27 


it)555o 


16-76 


894450I 44 


17 


io3o37 


i6-45 


996482 


•27 


io6556 


16-72 


8 9 3444| 43 


18 


104025 


16-41 


996465 


•27 


107559 


16-69 


892441 42 


19 


io5oio 


i6-38 


996449 


•27 


io856o 


i6-65 


891440J 41 


20 


105992 


i6-34 


996433 


• 27 


109559 


16-61 


890441 1 40 


21 


9*106973 


i6-3o 


9-9964I7 


•27 


9- no556 


i6-58 


10-8894441 39 


22 


107951 


16-27 


996400 


•27 


1 1 1 55i 


i6-54 


8884491 38 


23 


108927 


16-23 


996.384 


•27 


1 1 2543 


i6-5o 


8874571 3 7 


24 


109901 


16-19 


9 9 6368 


• 27 


n3533 


16-46 


886467! 36 


25 


110J73 


16- 16 


99&35I 


•27 


H452I 


i6-43 1 885479! 35 


26 


1 1 1842 


16- 12 


996335 


•27 


n55o7 


16-39 884493 j 34, 


27 


1 1 2809 


16-08 


9 9 63 1 8 


•27 


116491 


16 -36 | 8835o9| 33 


28 


113774 


i6-o5 


99&3o2 


.28 


117472 


16-32 


8825281 32 


29 


114737 


16-01 


996285 


.28 


1 i8452 


16-29 


88i548i 3 1 


3o 


115698 


i5-97 


996269 


• 28 


1 19429 


l6-2D 


88057 11 3o 


3i 


9-ii6656 


i5-q4 


9-996252 


.28 


9. 120404 


16-22 


10-879596; 29 


32 


117613 


15.90 


996235 


•23 


121377 


16-18 


878623 ; 28 


33 


118J67 


15.87 


996219 


.28 


122348 


i6-i5 


877662: 27 


34 


119519 


i5-83 


996202 


• 28 


123317 


16-11 


876683. 26 


35 


120469 


i5-8o- 


996185 


.28 


124284 


16-07 


875716; 25 


36 


1 2 14.17 


i5-t6 


996168 


•23 


125249 


16-04 


8747011 24 


37 


122362 


i5-73 ' 


996 1 5 1 


•23 


126211 


16-01 


873789^ 23 


3S 


l233o6 


15-69 


996134 


.28 


127172 


»5-97 


872828 22 


3 9 


124248 


i5-66 ' 


9951 17 


• 28 


128i3o 


i5-94 


871870 21 


4o 


125187 


i5-62 


996100 


.28 


129087 


i5 -oi 


870913 20 


4i 


9«i26i25 


15.D9 


9.996083 


.29 


9"i3oo4i 


i5-8 7 


10-869959: 19 
869006 18 


42 


127060 


1 5- 56 


996066 


.29 


1 30994 


i5-84 


43 


127993 


i5-52 


996049 


.29 


1 3 1944 


i5-8i 


868o56; 17 


44 


128923 


i5-49 


996032 


.29 


132893 


15.77 


8671071 16 


45 


129804 


1D-45 


996015 


•29 


i3383 9 


15-74 


866161! i5 


46 


130781 


i5-42 


993998 


.29 


134784 


i5-7i 


865216! 14 


47 


1 31706 


i5-3g 


993980 


•29 


135726 


15.67 


8642741 1 3 


48 


i3263o 


i5-35 


995963 


• 29 


136667 


15-64 


8633331 12 


49 


i3355i 


i5-32 


995946 


.29 


137605 


i5-6i 


862395 1 11 
861408! 10 


5o 


134470 


15.29 


995928 


.29 


138542 


i5-58 


5i 


9 135387 


i5-25 


9-99301 1 
990894 


.29 


9-139476 


i5-55 


io-86o524! 9 
85 9 5 9 ij 8 


52 


i363o3 


1 >• 22 


•29 


1 40409 


i5-5i 


53 


137216 
1 38 128 


i5- 19 


995876 


• 291 


i4i34o 


15.48 


85866oi 7 


54 


i5-i6 


995859 


•29 


142269 


i5-45 


85 77 3ii 6 


55 


139037 


I5-I2 


995841 


• 29 


143196 


1 5-42 


8568o4i 5 


56 


139944 


i5-09 


995823 


•29 


144121 


i5-39 


8558791 4 


57 


140800 


i5-o6 


995806 


.29 


143044, 


l5-35 


854956, 3 


58 


i4h54 


i5-o3 


995788 


.29 


140966 

146885, 


i5-3a 


854o34| 2 


5 9 


142655 


'i5-oo 


995771 


• 29! 


i5-29 


853n5 1 


60 


143555 


14-96 


995753 


• 29 


147803 1 


i5-26 


862197 


1 


CnBillb 


I). 


Sine 


S2°l 


Co tang. 1 


D. 


Tang. 


*L1 



26 


(8 


DEGREE 


=.) A TABLE 


OF LOGARITHMIC 




~M. 1 "Sane 


L>. 


Cosine 


D. 


Tang. 


D. 


Cotanj?. 




o 9- 143555 


14-96 


9-995753 


•3o 


9- 147803 


15-26 


io-852i97 

85l2§2 


6e 


1 


144453 


14-93 


995735 


•3o 


148718 


i5 


23 


£ 


2 


145349 


14-9° 


995717 


-3o 


149632 


i5 


20 


85o368 


3 


146243 


14-87 


995699 


•3o 


i5o544 


i5 


17 


849456 


U 


4 


I47i36 


14-84 


995681 


•3o 


i5i454 


i5 


14 


848546 


5 


148026 


14-81 


995664 


•3o 


i52363 


i5 


11 


847637 


55 


6 


148915 


14-78 


995646 


•3o 


153269 


i5 


08 


846 7 3i 


54 


7 


149802 


U-75 


995628 


•3o 


i54«74 


i5 


o5 


845826 


53 


8 


i5o686 


U-72 


995610 


•3o 


155077 


i5 


02 


844923 


52 


9 


i5i569 


14-69 


995591 


•3o 


155978 


14 


99 


844022 


5i 


10 


i5245i 


14-66 


995573 


•3o 


1 568 77 


14 


96 


843i23 


5o 


11 


9-i5333o 


14-63 


9-995555 


•3o 


9-157775 


14 


93 


10-842225 


49 


12 


1 54208 


14-60 


99 5 53 7 


•3o 


1 586 7 i 


14 


s 


841329 


48 


i3 


i55o83 


i4-5 7 


995519 


■ 3o 


. 1 5g565 


14 


840433 


47 


14 


155957 
1 5683o 


14-54 


9955oi 


-3. 


160457 


14 


84 


83 9 543 


46 


i5 


i4-5i 


995482 


• 3i 


i6i347 


T4 


81 


838653 


45 


16 


157700 


U-48 


993464 


-3i 


162236 


14 


79 


83 77 64 


44 


n 


1 58569 


14-45 


995446 


-3i 


i63i23 


14 


76 


8368 7 7 


43 


18 


159435 


14-42 


995427 


-3i 


164008 


• 4 


73 


835992 


42 


■9 


i6o3oi 


i4-3 9 


995409 


-3i 


164892 


14 


70 


835 1 08 


41 


20 


161164 


14-36 


995390 


•3i 


165774 


14 


67 


834226 


4o 


21 


9-162020 


14-33 


9-995372 


-3i 


9-i66654 


14 


64 


io- 833346 


ll 


22 


162885 


i4-3o 


995353 


-3i 


167532 


14 


61 


832468 


23 


163743 


14-27 


995334 


■ 3i 


168409 


14 


53 


83i5 9 i 


37 


24 


164600 


14-24 


9953i6 


■ 3i 


169284 


14 


55 


83o 7 i6 


36 


25 


165454 


14-22 


995297 


-3i 


170157 


14 


53 


829843 


35 


26 


166307 


14-19 


995278 


• 3i 


171029 


14 


5o 


828971 


34 


27 


167159 


14-16 


99J260 


-3i 


171899 


14 


47 


828101 


33 


23 


168008 


!4-i3 


995241 


-32 


172767 


14 


4i 


827233 


32 


<*9 


168856 


14-10 


990222 


-32 


173634 


14 


[2 


826366 


3 1 


3o 


169702 


14-07 


99j2o3 


•32 


174499 


14 


3 9 


8a55oi 


3o 


3i 


9-I70547 


i4-o5 


9-995184 


-32 


9-173362 


14 


35 


40-82 {63S 


29 


32 


i 7 i3J 9 


14-02 


993165 


-32 


176224 


14 


33 


823776 


28 


33 


172230 


13-99 


993146 


-32 


177084 


14 


3i 


822916 


27 


34 


173070 


l3- 9 6 


995127 


-32 


177942 


14 


28 


822058 


26 


35 


173908 


i3-94 


9g5io8 


-32 


173799 
i 79 655 


* i4 


25 


82I20I J 


25 


36 


174744 


13-01 


995089 


-32 


14 


23 


820345 


24 


u 


i 7 5j 7 8 


13-88 


995070 


-32 


i3o5o8 


14 


20 


819492 


23 


176411 


13-86 


993061 


-32 


i3i36o 


14 


17 


818640 


22 


39 


177242 


13-83 


Qg5o32 


-32 


182211 


14 


i5 


817789 


21 


40 


178072 


i3-8o 


9g5oi3 


-32 


1 33 059 


14 


12 


8 1 694 1 


20 


41 


9-178900 


13-77 


9.994993 


.32 


9- 183907 


14 


09 


10-8160931 19 


42 


179726 


13-74 


994974 


-32 


184752 


14 


07 


8i5248| 18 


43 


i8o55i 


13.72 


994933 


.32 


1 3 35 97 
186439 


14 


04 


8i44o3 n 


44 


i8i3 7 4 


i3-6 9 


994935 


•32 


14 


02 


8i356i 16 


45 


182196 


13-66 


994916 


.33 


187280 


i3 


99 


812720! i5 


46 


i83oi6 


13-64 


994896 


.33 


188120 


i3 


9 5 


81 1880 14 


47 


183834 


i3-6i 


994877 


.33 


i83 9 58 


i3 


93 


81 1042 i3 


43 


13465 i 


i3-5 9 


994857 


.33 


189794 


i3 


9 1 


810206 12 


49 180466 


13-56 


994838 


.33 


193629 


13 


89 


809371 11 


5o 


180280 


13-53 


994818 


.33 


191462 


i3 


86 


8o8538 10 


5i 


9-187092 


1 3 - 3 1 


9-994798 


.33 


9-192294 


|3 


84 


10-807^06! 9 
806876, 8 


52 


187903 


13-48 


994779 
994759 
994739 


.33 


1 93 1 24 


i3 


81 


53 
54 


188712 
189319 


13-46 
13-43 


.33 

• 33 


193953 
194780 


i3 
.3 


79 
76 


806047 : 7 

8o522o| 6 


55 


190325 


i3-4i 


994719 


.33 


195606 


1 3 


n 


804394! 5 


56 


191 i3o 


i3-38 


994700 


.33 


196430 


i3 


71 


803570J 4 


57 


191933 


i3-36 


994680 


.33 


197253 


i3 


69 


802747 3 


58 


102734 


1 3 - 33 


994660 


.33 


198074 


i3 


66 


801926 


2 


59 193534 


i3-3o 


994640 


.33 


198894 


i3 


64 


801 106 


I 


60 


194332 


i3-28 


994620 


.33 


199713 


,3 


Oi 


800287 


O 




Cosine 


L>. 


■Sine 


Cotanj?. 


D. 


Tano, 


Uk. 





SINES 


AND TANGENTS 


. (9 degrees/ 




27 


to. 


Sine 


D. 


Cosine 1 D. 


Tang. 


D. 


Cotang. 


60 


o 


9.194332 


i3 


28 


9-994620 


33 


9-199713 


i3 


61 


10-800287 


1 


193129 


i3 


26 


994600 




33 


200529 


1 3 


i? 


79947' 


It 


2 


190920 


i3 


23 


994580 




33 


2oi345 


i3 


56 


798655 


3 


196719 


i3 


21 


994360 




34 


202159 


i3 


J 4 


797841 


ii 


4 


197511 


i3 


l8 


994540 




34 


202971 


i3 


52 


797020 


56 


5 


io83o2 


i3 


16 


994319 




34 


203782 


i3 


49 


796218, 55 


6 


1 9909 1 


i3 


i3 


994499 




34 


204592 


i3 


47 


795408 54 


I 


199879 


i3 


1 1 


994479 




34 


205400 


i3 


45 


794600' 53 


200666 


i3 


08 


994459 




34 


206207 


i3 


42 


793793 j 52 


9 


20I43I 


i3 


06 


994438 




34 


207013 


i3 


4o 


792987 


5i 


IO 


202234 


i3 


04 


994418 




34 


207817 


i3 


38 


792183 


5o 


U 


9-203017 


i3 


01 


9-994397 




34 


9.208619 


i3 


35 


io-79i38i 


49 


12 


203797 


12 


99 


994377 




34 


209420 


i3 


33 


790580 48 


i3 


204377 


12 


96 


994357 




34 


210220 


13 


3i 


789780 


47 


U 


205354 


12 


94 


994336 




34 


21 1018 


i3 


28 


788982 


46 


ID 


2o6i3i 


12 


t 


9943i6 




34 


2ii8i5 


1 3 


26 


788i85 


45 


16 


206906 


12 


994295 




34 


212611 


i3 


24 


787389 


44 


17 


207679 


J2 


87 


994274 




35 


2i34o5 


i3 


21 


786595 


43 


18 


208432 


12 


85 


994254 




35 


214198 


i3 


19 


785802 


42 


*9 


209222 


12 


82 


994233 




35 


2U989 


i3 


17 


785oi 1 


41 


20 


209992 


12 


80 


994212 




35 


215780 


i3 


i5 


784220 


40 


21 


9-210760 


12 


78 


9-994191 




35 


9 . 2 i6568 


1 3 


12 


iG-783432 


ig 


22 


211626 


12 


75 


9941 7 1 




35 


217356 


i3 


10 


782644 


23 


212291 
2i3o55 


12 


73 


994160 




35 


218142 


i3 


08 


781858 


37 


24 


12 


7i 


994129 
994108 




35 


218926 


i3 


o5 


781074 


36 


25 


2i38i8 


12 


68 




35 


219710 


i3 


o3 


780290 


35 


26 


214379 
2i5338 


12 


66 


994087 




35 


220492 


i3 


01 


779508 


34 


2 


12 


64 


994066 




35 


221272 


12 


99 


778728I 33 


216097 
216854 


12 


61 


' 994045 




35 


222062 


12 


Q7 


777948I 32 


29 


12 


5 9 


994024 




35 


222830 


12 


94 


7771701 3i 


3o 


217609 


12 


5 7 


994oo3 




35 


2236o6 


12 


92 


776394I 3o 


3i 


9-2i8363 


12 


55 


9-993981 




35 


9-224382 


12 


90 


10--775618J 29 


32 


219116 


12 


53 


993960 




35 


225i56 


12 


88 


774844 


28 


33 


219868 


12 


5o 


993939 
993918 




35 


225929 


12 


86 


774071 


27 


34 


220618 


12 


48 




35 


226700 


12 


84 


7733oo 


26 


35 


221367 


12 


46 


993896 




36 


227471 


12 


81 


772529 25 


36 


222Il5 


12 


44 


993875 




36 


228239 


12 


79 


77'7 6 ii 24 


n 


222861 


12 


42 


993854 




36 


229007 


12 


77 


770993I 23 


2236o6 


12 


3 9 


993832 




36 


229773 


12 


73 


770227 22 


3 9 


224349 


12 


37 


99381 1 




36 


23o539 


12 


73 


769461 ! 21 1 


4o 


225092 

9-225833 


12 


35 


993789 
9-993768 




30 


23l3o2 


12 


7' 


768698 
10-767935 


20 | 


4i 


12 


33 




36 


9-232o65 


12 


69 


:? 


42 


226573 


12 


3i 


993746 




3o 


232826 


12 


67 


767174 


43 


22731 1 


12 


28 


993725 




36 


233586 


12 


65 


766414 


n 


44 


228048 


12 


26 


993703 




36 


234345 


12 


62 


765655 


16 


45 


228784 


12 


24 


993681 




36 


235io3 


12 


60 


764897 


i5 


46 


229618 


12 


22 


993660 




36 


235859 


12 


58 


764141 


14 


47 


230252 


12 


20 


99 3638 




36 


2366i4 


12 


56 


763386 


i3 


48 


230984 


12 


18 


993616 




36 


237368 


12 


54 


762632 


12 


49 


23l7U 


12 


16 


993594 




37 


238i2o 


12 


52 


761880 


1 1 


5o 


232444 


12 


14 


993572 




37 


238872 


12 


5o 


761128 


10 


5i 


9'233i72 


12 


12 


9'99355o 




37 


9.239622 


12 


48 


10-760378 


I 


52 


233899 
234625 


12 


09 


993528 




37 


240371 


12 


46 


739629 


53 


12 


07 


9935o6 




37 


241 1 18 


12 


44 


758882 


7 


54 


235349 
236073 


12 


o5 


993484 




37 


24i865 


12 


42 


758i35 


6 


55 


12 


o3 


993462 




37 


242610 


12 


40 


757390 


5 


56 


236795 
2375i5 


12 


01 


993440 




37 


243354 


12 


38 


756646 


4 


ll 


II 


99 


993418 




37 


244097 
• 24483 9 


12 


36 


755903 


3 


58 


238235 


II 


97 


993396 




37 


12 


34 


755i6i 


2 


5<) 


238953 


II 


9 5 


993374 




37 


245579 


12 


32 


734421 


1 


6o 


239670 


II -93 


99335i 


•3 7 


246319 


12 


3o 


75368i 







Cosine 


D. 


Sine 


80° 


Cotang. 


D. 


Taner. 


M., 



12 



28 


(10 


DEGREES.) A 


TABLE OF LOGARITHMIC 




M. 


Sine 


D. 


Cosine 


D. 


Tang. 


r>. 


Cotang. 







9-239670 


il. 9 3 


9-99335i 


.37 


9 -2463 19 


12'3o 


io^-^5368T 


60 


i 


24o386 


1I.Q1 


993329 


.37 


247057 


12-28 


752943 


5 9 


2 


241 101 


II.89 


993307 


.37 


247794 


12-26 


7522o6| 58 


3 


241814 


II.87 


99 3285 


.37 


24853o 


12-24 


75i470! 57 


4 


242526 


n-85 


993262 


.37 


249264 


12-22 


75o 7 36! 56 


5 


243237 


11-83 


993240 


ii 


24999 8 


12-20 


75ooo2| 55 


b 


243947 


11. 81 


993217 


25o73o 


12-18 


7492701 54 
. 74853 9 53 


I 


244656 


11-79 


993195 


•38 


25i46i 


12-17 


245363 


H-77 


993172 


•38 


252191 


I2-l5 


747809 52 


9 


246069 


11-73 


993149 


• 38 


252920 


I2-I3 


747080' 5i 


10 


246773 


II-73 


993127 


• 38 


253648 


12- II 


7463521 5o 


ii 


9-247478 


II- 71 


9.993104 


•38 


9-254374 


12 09 


10-7456261 40 
744900! 48 


12 


248181 


11.69 


993081 


• 38 


255ioo 


12-07 


i3 


248883 


11.67 
n-65 


993059 


•38 


255824 


12 03 


744176' 47 


14 


249583 


9g3o36 


• 38 


256547 


12-03 


743453! 46 


i5 


250282 


n-63 


993oi3 


• 38 


257269 


12-01 


742731 45 


16 


250980 


n-6i 


992990 


• 38 


257990 


12-00 


742010! 44 


\l 


251677 


wit 


992967 


• 38 


258710 


II.98 


7412901 43 


232373 


992Q44 


• 38 


259429 


II.96 


740571 42 


19 


253067 


n-56 


992021 


• 38 


260146 


u-94 


739854 4i 


20 


253761 


n-54 


992898 


• 38 


26o863 


11-92 


739137 40 

10-738422 39 

73 77 o8| 38 


21 


0-254453 


11-52 


9-992875 


• 38 


9-261578 


\\T 9 


22 ' 255i44 


11 5o 


992852 


• 38 


262292 


23 | 255834 


11.48 


992829 


19 


263oo5 


n.87 


736995! 37 


24 


256523 


11.46 


992806 


•?9 


263717 


n-85 


7 36233i 36 


23 


25721 1 


n-44 


992783 


•59 


264428 


n-83 


735572 


35 


26 


257898 


11-42 


992759 


I' 


2 65i38 


n-8i 


734862 


34 


2 


258583 


11. 4i 


992736 


•?9 


265847 
266555 


W.ll 


734i53 


33 


259268 


11.39 


992713 


19 


733445 


32 


29 


259951 


11.37 


992690 


•?9 


267261 


11.76 


732 7 3o 
732o33 


3i 


3o 


26o633 


n-35 


992666 


• 3 9 


267967 


n-74 


3o 


3i 


9- 2613 14 


n-33 


9-992643 


•3o 


9-268671 


11-72 


io>73i329 
730623 


ll 


32 


261994 


ii-3i 


992619 


•39 


269375 


11-70 


33 


262673 


11. 3o 


992596 


•39 


270077 


11.69 


729923 


27 


34 


26335i 


11-28 


992572 


■39 


270779 


11-67 


729221 


26 


35 


264027 
26470J 


11-26 


992 549 


•?9 


271479 
272178 


1 1. 65 


728521 


25 


36 


11-24 


992525 


• 3 9 


n-64 


727822 


24 


ll 


265377 


11-22 


992 5o 1 


• 3 9 


272876 


11-62 


727124 


23 


266o5i 


11-20 


992478 


.40 


273573 


11. 60 


726427' 22 


3 9 


266723 


II- 19 


992454 


.40 


274269 


n-58 


725731 1 21 


40 


267395 


II- 17 


992430 


.40 


274964 


w.n 


725o36 20 


4i n.263o65 


11 • i5 


9-992406 


.40 


9-275658 


10-724342: 19 


42 


268734 


11. i3 


992382 


.40 


276351 


11.53 


723649 18 


43 


269402 


1 1 - 1 1 


992359 


.40 


2-' -7043 


11. 5i 


722957 17 
722266! 16 


A\ 


270069 


11. 10 


992333 


.40 


277734 


ii-5o 


45 


270733 


11.08 


992311 


.40 


278424 


11.48 


721576 


13 


46 


271400 


11. c6 


992287 


.40 


279113 


1 1 .43 


720887 


14 


A \ 


272064 


11. o5 


992263 


.40 


279801 


720199 


i3 


48 


272726 


11. o3 


992239 


■ 40 


280488 


n-43 


719912 


12 


^ 9 


273388 


II -CI 


992214 


.40 


281174 


H-41 


718826 


II 


5o 


274049 


10-99 


992190 


.40 


28i853 


ii-4o 


718142 


IO 


5i g 


10-98 


9-992166 


.40 


9-282542 


n-38 


10 717458 


I 


52 


275367 


10-96 


992142 


.40 


283225 


n-36 


716775 


53 


276024 


• io-9-i 


9921 17 


•41 


283907 


n-35 


7 1 6093 


7 


54 


276681 


10-92 


992093 


•41 


084588 


n-33 


70412 


6 


55 


277337 


1 • 9 1 


992069 


•41 


285268 


n-3i 


714732 


5 


56 


277991 


10- 89 


992044 


•41 


285947 


n-3o 


7i4o53 4 


57 


278644 


10-87 


992020 


•41 


286624 


11-28 


7i33 7 6 3 


58 


279297 


io-86 


991996 


•41 


•287301 


11-26 


7 1 2 699 J 2 


J 9 


279948 


10-84 


991971 


•41 


287977 
288652 


11-25 


71 2023 1 1 


60 


280099 


10-82 


991947 


•41 


1123 


7113481 


',_. 


Cosir.a 


D. 


Sine 


790 


Couuiy. 


D7~ 


_Taug.._ 


_.M t 



SINES AND TANGENTS. (11 DEGREES.) 



29 



FTC 


| Sine 


D. 


Cosine | D- 


I Tang. 


D. 


Cotaner. f 


1 ° 


9 -280599 


10-82 


9-991947 


•41 


9-288652 


11-23 


10-711348 60 


1 


281248 


10. 81 


991922 


•4i 


289326 


11-22 


710674 5o 
7100011 58 


2 


281897 


10-79 


991897 


•4i 


289999 


11-20 


3 


282544 


10-77 


991873 


•41 


290671 


II-l8 


709329! 57 


4 


283190 


10-76 


991848 


•4i 


291342 


11-17 


7o8658 56 


5 


283836 


10-74 


991823 


•4i 


29201J 


n-i5 


707987 


55 


6 


284480 


10-72 


991799 


•41 


292682 


11. 14 


707318 


54 


I 


285i24 


10-71 


99H74 


•42 


293350 


11-12 


7o665o 


53 


285 7 66 


10-69 


991749 


•42 


294017 


II -II 


705983 


52 


9 


286408 


10-67 


991724 


•42 


294684 


11-09 


7o53i6 


5i 


IO 


287048 


io-66 


991699 


•42 


295349 


H-07 


70465 1 


5o 


ii 


9.287687 


10-64 


9-991674 


•42 


9-296013 


11 -06 


10-703987 


49 


12 


288326 


io-63 


991649 


•42 


296677 


li-o4 


7o3323 


48 


i3 


288964 


io-6i 


991624 


•42 


297339 


n-o3 


702661 


47 


U 


289600 


10-59 


991599 


•42 


298001 


II-OI 


701999 


46 


ID 


2go236 


10-58 


99 l5 74 


•42 


298662 


11-00 


7oi338 


45 


16 


290870 


io-56 


99i549 


•42 


299322 


10-98 


• 700678 


44 


n 


291504 


io-54 


991524 


•42 


299980 


10-96 


700020 


43 


18 


292137 


io-53 


991498 


.42 


3oo638 


10-90 


699362 


42 


'9 


292768 


io-5i 


991473 


•42 


301295 


10-93 


698705 


4i 


20 


293399 


io-5o 


99U48 


•42 


3oi95i 


10-92 


698049 


40 


21 


9-294029 


10-48 


9-991422 


•42 


9-302607 


10-90 


10-697393 


39 


22 


294658 


10-46 


991397 


•42 


3o326i 


10-89 


696739 


38 


23 


293286 


io-45 


991372 


•43 


3o3ni4 
304067 


10-87 


696086 


37 


24 


29D913 
296539 


io-43 


991346 


•43 


io-86 


695433 


36 


20 


10-42 


991321 «43 


3o52i8 


10-84 


694782 


35 


26 


297164 


10-40 


991295 *43 


3o586 9 


io-83 


694131 


34 


27 


297788 


10-39 


991270I «43 


3o65i9 
307168 


io-8i 


693481 


33 


28 


298412 


io-37 


991244! "43 


io-8o 


692832 


32 


29 


299034 


io-36 


991218I -43 


3o 7 8i 5 


10-78 


6 9 2i85 


3i 


3o 
3i 

32 


299655 


io-34 


991 193 


•43 


3o8463 


10-77 


691537 


3o 


9-300276 


10-32 


9-991167 


•43 


9-309109 


10*75 


10-690891 


29 


300895 


io-3i 


99n4i 


•43 


309754 


10-74 


690246 


28 


33 


3oi5i4 


I0-2O 


9911 i5 


•43 


310398 


10-73 


689602 


27 


34 


302132 


10-28 


991090 


•43 


3no42 


10-71 


688958 


26 


35 


302748 


10-26 


991064! -43 


3u685 


10-70 


6883 1 5 


25 


36 


3o3364 


10-25 


99io38 -43 


312327 


10.68 


687673 


24 


u 

3 9 


303979 


10-23 


991012 -43 


312967 


10-67 


68 7 o33 


23 


3o4393 


10-22 


990986 


• 43 


3i36o8 


10-65 


6863 9 2 


22 


3o5207 


10-20 


990960 


•43 


3i4247 


10-64 


685753 


21 


4o 


3o58i9 


io- 19 


990934 


•44 


3 1 4885 


10-62 


680110 


20 


4i 


9-3o643o 


10-17 


9-990908 


•44 


9«3i5523 


10-61 


10-684477 


3 


42 


307041 


IO-IO 


990882 


•44 


3i6 1 59 

316795 
3 n43o 


10-60 


683841 


43 


307600 


10-14 


99o855 


•44 


10-58 


6832o5 


17 


44 


308209 


10; 13 


990829 


•44 


io-57 


682570 


16 


45 


308867 


lO-II 


990803 


•44 


318064 


io-55 


68i 9 36 


i5 


46 


309474 


IO-IO 


990777 


•44 


3i86 97 


io-54 


68i3o3 


14 


47 
48 


3 1 0080 


10-08 


990750 


•44 


319329 


io-53 


680671 


i3 


3io685 


10-07 


990724 


•44 


3 1 996 1 


io-5i 


680039 


12 


49 


311289 


io-o5 


990697 


•44 


320592 


io-5o " 


679408 


11 


00 

5i 

52 

53 
54 
55 
56 

58 


311893 


10-04 


990671 


•44 


321222 


10-48 


678778 


10 , 


9-312495 


io-o3 


9-990644 


•44 


9-32i85i 


io-47 


10-678149 


8 


3 1 3097 


10-01 


990618 


•44 


322479 


io-45 


677521 


3 1 36 9 8 


10-00 


990591! .44 


323io6 


io-44 


676894 


7 


314297 


9-98 


990060 .44 


323 7 33 


10-43 


676267 


6 


314897 


9-97 


99o538 «44 


324358 


10-41 


675642 


5 


3i5495 


9-96 


99o5n -45 


324983 


10-40 


675017 


4 


316092 
3i668 9 


9.94 


99o485| .45 


325607 


10-39 


674393 


3 


9-93 


990458 -45 


32623i 


10-37 


673769 


2 


& 


317284 


9-91 


990431 ^45 


326853 


io-36 


673147 


1 


_6o_ 


317879 


9-9° 


990404] «45 


327470 


io-35 


672525 





Cosine 


D. 


Sine i78° 


Cotang. 


r>. 


Tang. 


M. 



30 



(12 DEGREES.) A TABLE OF LOGARITHMIC 



M. 




1 Sine 


D. 


Cosine | D. 1 Tang. 


1 D. 


Cotang 


1 1 


9-317879 


?s 


9-990404 -45 9-327474 io-35 


10-672526' 60" ! 
671905 1 5o 
671285; 58 


i 


3i8473 


990378' .45 328095 io-33 


2 


3 1 9066 


9-87 


99o35i .45 32871: 


i 10-32 


3 


319658 


9-86 


99o32zj 


•45 32933/ 


. io-3o 


670666! 57 


4 


320249 


9-84 


99020- 


-At 


32995I 


10-29 


670047 56 


5 


320840 


9-83 


99027c 


•At 


> 33037C 


10 28 


66g43a 


55 


6 


32i43o 


9.82 


9902^ 


•At 


33 118- 


10-26 


6688 1 3 


54 


1 


322019 


9-80 


99021' 


•At 


33i8o3 


10-25 


668197 


53 


8 


322607 


9-79 


990188 


•45 


33 24 1 £ 


10-24 


66 7 58a 


5a 


9 


323io4 


9-77 


990161 


•45 333o33 


IO-23 


666967 
66635 4 


5i 


10 


323780 


9.76 


990134 


•45i 333646 


10-21 


5o 


ii 


9-324366 


9-75 


9-990107 


•46] 9-334259 


10-20 


10-665741 


49 


12 


32495o 


9- 7 3 


990079 


.46 334871 


IO-I9 


665 1 29 
6645i8 


48 


s3 


325534 


9-72 


990052 


.46! 335482 


10-17 


47 


14 


326117 


9-70 


990025 


•46 


33609c 


io- 16 


663907 


46 


i5 


326700 


9-69 
9-68 


989997 


•46 


336702 


io- 15 


663 298 
662689 


45 


16 


327281 


989970 


.46 


3373 1 1 


io- 13 


44 


\l 


327862 


9-66 


989942 


• 46 


337919 


IO-I2 


662081 


43 


328442 


9-65 


9S9915 


•46 


338D27 


IO-II 


661473 


42 


»9 


329021 


9.64 


9S9887 


•46 


339i33 


10- 10 


660867 


41 


20 


329599 


9-62 


989860 


•46 


339739 


10-08 


660261 


40 


21 


9-330176 


9-61 


9-9S 9 832 
909804 


.46 


9- 34o344 


10-07 


10-659656 


\l 


22 


33o753 


9-60 


•46 


340948 
34i552 


io- 06 


659052 


23 


33i3: 9 


9-58 


989777 


.46 


io-o4 


658448 


3 7 


24 


33190J 


9-57 


989749 


•47 


342i55 


10 -o3 


65 7 845 


36 


25 


332478 


9-56 


989721 


•47 


342757 


10-02 


657243 


35 


26 


333o5i 


9-54 


989693 


•47 


343358 


10-00 


656642 


34 


27 


333624 


9-53 


989660 


•47 


343958 
344558 


9-99 
9-98 


656o42 


33 


28 


334iq5 


9-52 


989637 


•47 


65544s 


32 


29 


334766 


9-5o 


989609 


•47 


345i 57 


9-97 


654843 


3 1 


3q 


335337 


9-49 


9 8 9 58 2 


•47 


345755 


9-96 


654245 


3o 


3i 


9-335906 


9-48 


9-989553 


•47 


9-346353 


9-94 


io-653647 


3 


32 


336475 


9-46 


989525 


•47 


346949 
347545 


9-93 


653o5i 


33 


337043 


9.45 


989497 


•47 


9-92 


652455 


27 


34 


337610 


9.44 


989469 


•47 


348141 


9-91 


65i85 9 


26 


35 


338i 7 6 


9.43 


989441 


-.47 


348735 


9-00 


65i265 


25 


36 


338742 


9-4i 


989413 


•47 


349329 


9-88 


650671 


24 


u 


33g3o6 


9-4o 


9 8 9 3S4 


•47 


349922 
35o5i4 


9-87 


650078 


23 


339871 


9.39 


9893 56 


•47 


9-86 


649486 


22 


39 


340434 


9.37 


989328 .47 


35no6 


9-85 


648894 


21 


40 


340996 


9-36 


989300 


•47 


351697 


9-83 


6483o3 


20 


41 


9-34i5a8 


9-35 


9.989271 


•47 


9.352287 


9-82 


10-647713 


\l 


42 


342119 


9-34 


989243 


•47 


352876 


9-81 


647124 


43 


342679 


9-32 


989214 


•47 


353465 


9-80 


6465351 17 
645947 16 


44 


343239 


9-3i 


989186 


•47 


354oo3 


9-79 


45 


343797 


9«3o 


9891 5 7 


•47 


354640 


9-77 


645360! 1 5 


46 


344335 


9-29 


989128 


•48 


355227 


9-76 


644773 


14 


h« 


344912 


9-27 


989100 


•48 


3558i3 


9-75 


644187 


i3 


345469 


9-26 


989071 


-48 


3563 9 8 


9.74 


643602 


12 


49 


346024 


9-25 


989042 


-48 


356982 
35 7 566 


9.73 


643oi8 


1 1 


5o 


346579 


9-24 


989014 
9-980985 


•48 


9-71 


642434 


10 


5i 


9-347134 


922 


-48 


9 -358 1 49 


9-70 


io-64i85i 




52 


347687 


9-21 


988956 


-48 


358 7 3i 


?a 


641269I £ 


53 


348240 


9-20 


988927 


•48 


35 9 3i3 


6406871 7 


54 


348792 


9.19 


988898 


• 48 


35 9 8 9 3 


9-67 


640107; 6 


55 


349343 


9-17 


988869 


.48 


36o474 


9 .66 


6395261 5 


56 


349893 


9- 16 


988840 


• 48 


36io53 


9-65 


638 Q 47 '4 


8 


35o443 


9- 15 


9888 n .49 


36i632 


9-63 


638368 3 


350992 
35i54o 


9-14 


9887821 .49 


362210 


9-62 


637790 2 


5 9 


9-i3 


988753! .40 


362787 


9-61 


63 7 2 1 3 1 1 


60 


352088 


9. 11 


988724 .49 363364| 


9-60 


636636 




Cosine 


D. 


Sine ?T°I Cotang. 1 


7~p. 


Tang, j 


MJ 





SIXES AND TANGENTS 


(13 DEGREES. 


) 


31 


M. 




Sine 1 D. 


Cosine 


D - 


Tang. 


I). 


Cotang. 


• 


9-352088 


9-n 


9-988724 


•49 


9-363364 


9-60 


10-636636 


60 


i 


352635 


9 


10 


988695 


■49 


36394o 


9 .5o 


636o6o 


5 9 


2 


353i8i 


9 


2 


988666 


•49 


3645 1 5 


9-58 


635485 


58 


3 


353726 


9 


9 88636 


•49 


365ooo 


9.57 


634910 


5 7 


4 


354271 


9 


07 


988607 


•49 


365664 


9.55 


634336 


56 


5 


3548i5 


9 


o5 


988578 


•49 


366237 


9-54 


633 7 63 


55 


6 


355358 


9 


04 


988548 


•49 


3668io 


9-53 


633190 


54 


7 


355qoi 


9 


o3 


988519 


•49 


36 7 382 


9-52 


6326i8 


53 


8 


356443 


9 


02 


988489 


•49 


367953 


9-5i 


632047 


52 


9 


356o84 
357524 


I 


01 


988460 


.49 


368524 


9-5o 


631476 


5i 


10 


99 


988430 


.49 


369094 


1% 


630906 


5o 


ii 


9.358064 


8 


98 


9.988401 


•49 


9-369663 


io-63o337 


% 


12 


3586o3 


8 


97 


988371 


•49 


370232 


9.46 


629768 


i3 


359141 


8 


96 


988342 


•49 


370799 


9-45 


629201 
62,8633 


47 


14 


359678 


8 


9 3 


9 883 1 2 


.50 


371367 


9.44 


46 


i5 


36o2i 5 


8 


9 3 


988282 


•5o 


3.71933 


9-43 


628067 


45 


16 


360752 


8 


92 


988252 


-50 


372499 


9-42 


627501 


44 


n 


361287 


8 


91 


988223 


-50 


373064 


9-41 


626936 


43 


18 


361822 


8 


£ 


988193 


•5o 


373629 
374193 


9-40 


626371 


42 


*9 


362356 


8 


9 88i63 


•5o 


9 .3o 


625807 


4i 


20 


362889 


8 


88 


9S8i33 


•5o 


3747^6 


9-38 


625244 


40 


21 


9-363422 


8 


87 


9-988103 


•So 


9-375319 


III 


10-624681 


3 2 

38 


22 


363 9 54 


8 


85 


988073 


•50 


37588i 


6241 19 
623558 


23 


364485 


8 


84 


988043 


•5o 


376442 


9-34 


37 


24 


365oi6 


8 


83 


9 88oi3 


• 5o 


377003 


9-33 


622997 
622437 


36 : 


25 


365546 


8 


82 


987983 


•50 


377563 


9-32 


35 


26 


366075 


8 


81 


987953 
987922 


■50 


378122 


9 .3i 


621878 


34 


27 


3666o4 


8 


80 


•5o 


378681 


9>3o 


621319 


33 


28 


367i3i 


8 


79 


987892 


-5o 


379239 


III ' 


620761 


32 


29 


367659 


8 


77 


987862 


-50 


379797 
38o354 


620203 


3i 


3o 


368i8d 


8 


76 


987832 


•5i 


9-27 


619646 


3o 


3i 


9-368711 


8 


7 5 


9.987801 


•5i 


9-380910 


9-26 


10-619090 
6i8534 


2 


32 


369236 


8 


74 


987771 


-51 


38i466 


9-25 


33 


369761 


8 


73 


987740 


•5i 


382020 


9-24 


617980 


27 


34 


3 7 o285 


8 


72 


9S7710 


.51 


3S 2 5 7 5 


9-23 


617425 


26 


35 


370808 


8 


7i 


9S7679 


.51 


383i29 


9-22 


616871 


25 


36 


37i33o 


8 


70 


987649 

987618 


.51 


383682 


9-21 


6 1 63 1 8 


24 


ll 


371S52 


8 


69 


• 51 


384234 


9-20 


615766 


23 


372373 


8 


67 


9 8 7 588 


.51 


384786 


9-19 


6i52i4 


22 


3 9 ' 


372894 


8 


66 


987557 


.51 


385337 


9. 18 


6i4663 


21 


4o 


373414 


8 


65 


987526 


•51 


385888 


9-17 


614112 


20 


4i 


9-373933 


8 


64 


9-987496 


.51 


9-386438 


9«i5 


io-6i3562 


1? 


42 


374452 


8 


63 


987465 


.51 


386987 


9-14 


6i3oi3 


18 


43 


374970 


8- 


62 


987434 


.51 


38 7 536 


9- 13 


612464 


H 


44 


373487 


8- 


61 


987403 


.52 


388o84 


9-12 


611916 


16 


45 


376003 


8- 


60 


987372 


.52 


38863i 


911 


61 1369 


i5 


46 


376019 


8 


ll 


98^341 


.52 


389178 


9-10 


^ 610822 
610276 


14 


47 


377033 


8- 


987310 


.52 


389724 


9.09 


i3 


48 


377549 


8- 


5 7 


987279 


.52 


390270 


9-08 


609730 


12 


i 9 


378063 


'8 


56 


987248 


• 52 


3go8 1 5 


9.07 


609 1 85 


II 


i° 


378577 


8- 


54 


987217 


.52 


391360 


9-06 


608640 


IC 


i l 


9-379089 


8 


53 


9-987186 


.52 


9-391903 


9>o5 


10-608097 


I 


\l 


379601 


8 


52 


987155 


•52 


392447 


9.04 


607553 


53 


38on3 


8 


5i 


987124 


.52 


392989 


9-o3 


607011 


7 


5* 


380624 


8 


5o 


987092 


.52 


39353i 


9-02 


606469 


6 


5o 


38n34 


8 


% 


987061 


.52 


394073 


9-01 


605927 


5 


56 


38i643 


8 


987030 


•52 


394614 


9 -00 


6o5386 


4 


^ 7 


382i5 2 


8 


47 


986998 


•52 


395154 


8-99 


604846 


3 


58 


382661 


8 


46 • 


9S6967 


•52 


395694 


8-98 


6o43o6 


2 


5 9 


383 1 68 


8- 


45 


986930 


•52 


396233 


8-Q7 


603767 


1 


6o 


3836 7 5 
Cosine 


8-44 


986904 


•52 


396771 


8-96 


6o32'j9 





D. 


Sine 


76° 


Cotang. 


D. 


Tang 


TTJ 



82 


H 


4 DEGREES.) A 


TABLE OF LOGARITHMIC 




M. 


Sine 


D. 


Cosine 


D. | Tang. 


D. 


Cotang. 







Q-3836 7 * 


8-44 


9-986904 


•52 


9.396771 


8- 9 6 


io-6o322c; 


60 


i 


384i82 


8.43 


986873 


•53 


397309 


8.96 


60260J 
602164 


5 -2 

58 


2 


38468- 


8-42 


986841 


■53 


397846 


8. 9 5 


3 


385io2 


8-4i 


986809 


• 53 


3 9 8383 


g-"94 


60161- 


57 


4 


3856 9 - 


8-40 


986778 


•53 


398919 


8. 9 3 


601081 


56 


! 5 


386201 


8-3 9 


986746 


•53 


399455 


8.92 


6oo545 


55 


6 


38670^ 


8-38 


986714 


• 53 


399990 


8-91 


600010 


54 


I 


387201 


8-3 7 


9 86683 


•53 


400024 


8-90 


699476 

598942 


53 


387709 


8-36 


986651 


•53 


4oio58 


8-89 
8-83 


52 


9 


3882ic 


8-35 


986619 


• 53 


401591 


598409 


5i 


10 


38871 1 


8-34 


986587 


•53 


402124 


8-87 


597876 


5o 


ii 


9-389211 


8-33 


9-986555 


•53 


9 -402656 


8-S6 


i0'5 9 7J44 


2 


12 


38971 1 


8-32 


9 865 2 3 


• 53 


403187 


8-85 


5 9 68i3 


i3 


390210 


8-3i 


986491 
986459 


•53 


403718 


8- 84 


596282 


47 


14 


390708 


8-3o 


•53 


404249 


8-C3 


595751 


46 


i5 


391206 


8-28 


86427 
986395 


•53 


404778 


8-82 


595222 


45 


16 


391703 


8-27 


•53 


4o53o3 


8-Si 


594692 


44 


17 


392199 
39269D 


8-26 


9 86363 


•54 


4o5336 


8-So 


594164 


43 


18 


8-25 


9 8633 1 


•54 


4o6364 


3 


5 9 3636 


42 


*9 


393191 
3 9 3685 


8.24 


986299 


•54 


406892 


593108 


41 


20 


8-23 


986266 


•54 


407419 


8.77 


592081 


40 


21 


9-394I79 
394673 


8-22 


9.986234 


•54 


9.407940 


8.76 


io«5 9 2o55 


39 


22 


8-21 


986202 


•54 


408471 


8. 7 5 


5 9 i52 9 


38 


23 


3 9 5i66 


8- 20 


986169 


•54 


408997 


8-74 


5 9 ioo3 


37 


24 


3 9 5658 


8-19 
8.18 


986137 


•54 


4og52i 


g-74 


590479 
589955 


36 


25 


396150 


986104 


•54 


4ioo45 


8- 7 3 


35 


26 


396641 


8.17 


986072 


•54 


4io56g 


f'" 2 


58 9 43 1 

588908 


34 


2 


397132 


8-17 


986039 


•54 


411092 


8.71 


33 


397621 


8- 16 


986007 


■54 


41161D 


8-70 


588385 


32 


£9 


3981 1 1 


8- 15 


985974 


•54 


412:37 


8-69 
8-68 


58 7 S63 


3i 


3o 


398600 


8.14 


985942 


•54 


412653 


587342 


3o 


3i 


9-399088 


8-i3 


9-985909 


• 55 


9^413179 


8-67 


10-586821 


20 


32 


399575 


8-12 


985876 


• 55 


413699 


8-66 


5863oi 


28 


33 


400062 


8-n 


985843 


• 55 


414219 


8-65 


585 7 8i 


27 


34 


4oo54o 
4oio35 


8-io 


9858i 1 


• 55 


414733 


8-64 


585262 


26 


35 


8-09 


9 85 77 8 


• 55 


41 5257 


8-64 


584743 


25 


36 


4oi52o 


8-o8 


985745 


• 55 


4i5775 


8-63 


584225 


24 


37 


4o2oo5 


8-07 


985712 


• 55 


416293 


8-62 


583707 


23 


38 


402489 


8-o6 


985679 


• 55 


416810 


8-6i 


5S3190 


22 


3 9 


402972 


8-o5 


9 85646 


• 55 


417326 


8- 60 - 


582674 


21 


4o 


4o3455 


8-04 


9 856i3 


• 55 


4i7 8 42 


8-5o 

8-53 


582i58 


20 


4i 


9-4o3938 


8-o3 


9.985580 


• 55 


9-4i3358 


10 081642 


10 


42 


404420 


8-02 


985547 


• 55 


4i83 7 3 


8.57 


581127 


l8 


43 


404901 


8-oi 


9855i4 


• 55 


419387 


8-56 


58o6i3 


17 


44 


4o5382 


8-oo 


985480 


• 55 


419901 


8.-55 


580099 
579585 


16 


45 


4o5862 


7-99 


985447 


• 55 


420415 


8-55 


i5 


46 


4o63^i 
406.820 


7-98 


985414 


• 56 


420927 


8-54 


579073 
578560 


14 


% 


7-97 


9 8538o 


• 56 


421440 


8-53 


i3 


407299 


7-96 


985347 


• 56 


421952 


8-02 


578048 


12 


4 9 


407777 


7- 9 5 


9853i4 


• 56 


422463 


8-5i 


577037 


11 


5o 


408254 


7-94 


985280 


• 56 


422974 


8-5o 


577026 


10 


\* 


9 408731 


7-94 


9-985247 


• 56 


9-423484 


8-4o 


10-576016 


1 


s 


409207 


7- 9 3 


9852i3 


•56 


423993 
424003 


8.48 


576007 


53 


409682 


7-92 


985180 


• 56 


b 48 


575497 


-i 


54 


410157 


7-91 


985146 


•56 


425on 


8.47 


574989 


6 


55 


4io632 


7-9° 


9 85n3 


•56 


4255i9 


8-46 


574481 


5 


56 


41 1 106 


7-89 


985079 
985045 


•56 


426027 


8-45 


573973; 


4 


57 


411579 


7-88 


'& 


426534 


8. 44 


573466, 


3 


58 


412052 


7-87 


980011 


•ii 


427041 


8-43 


572959 


2 


5 9 


412D24 


7-86 


984978 


•£ 


427547 


8-43 


572453! 


1 


6o 


412996 


7-85 


984944 


56 


428o52 


8-42 
I). 


5719431 
Tohg. 1 





L- 


Cosine 


J). 1 


Sine 1 


750 1 


Cotang. 


■MTj 





SINES AND TANGENTS. 


(15 DEGREES. 


) 


3? 


p: 


Sine 


D. 


Cosine j D. 


Tang. 


D. 


Cotang 1 
10-3719481 60 


o 


9.412996 


7-85 


9-984944! 


37 


9-428o52 


8 


•42 


I 


413467 


7 


84 


984910 


37 


428557 


8 


41 


571443 59 
5 7 o 9 38 58 


2 


4)3938 


7 


83 


984876* 


5 7 


429062 


8 


•40 


3 


414408 


7 


83 


984842! 


37 


429566 


8 


• 3o 


570434 57 


4 


414878 


7 


82 


984808' 


57 


430070 


8 


■38 


569930! 56 


5 


4i5347 


7 


81 


934774 


•57 


43o573 


8 


•33 


569427! 55 


6 


4i58i5 


7 


80 


984740 


•37 


431075 


8 


•37 


568925; 54 


I 


416283 


7 


79 


084706 




37 


43 1 577 


8 


36 


568423 


i 53 


416731 


7 


7 8 


984672 




•57 


432079 


8 


35 


667921 


52 


9 


417217 


7 


77 


984637 




•37 


43258o 


8 


34 


567420 


1 5l 


10 


417684 


7 


76 


984603 




-n 


433oSo 


8 


33 


566920 


5o 


i i 


9-41 81 5o 


7 


73 


9-984569 




37 


9'43358o 


8 


-32 


io- 366420 


49 


12 


4i86i5 


7 


74 


984535| 


5 7 


434o8o 


8 


•32 


565920 


48 


i3 


419079 


7 


73 


9845001 


57 


434579 


8 


3i 


565421 


47 


14 


419544 


7 


?3 


984466 




U 


435078 


8 


3o 


564922 


46 


13 


420007 


7 


7^ 


084432 




435576 


8 


29 


564424 


45 


16 


420470 


7 


71 


984397 




58 


436073 


8 


28 


563 9 27 


44 


l l 


420933 


7 


70 


984863 




58 


436570 


8 


28 


56343o 


43 


18 


421395 


7 


ft 


984328 




58 


437067 


8 


27 


562933 


42 


*9 


421837 
4223i8 


7 


984294 




58 


437563 


8 


26 


562437 


4i 


20 


7 


67 


984239 




58 


438o5 9 


8 


25 


56 1 941 


4o 


21 


9-422778 


7 


67 


9-984224 




58 


9-438554 


8 


24 


10.561446 


3 9 


22 


423238 


7 


66 


984190 




58 


439048 


8 


23 


560952 


38 


23 


42 ^697 


7 


65 


984155 




58 


439543 


8 


23 


56o457 


37 


24 


424136 


7 


64 


984120 




58 


44oo36 


8 


22 


559964 


36 


23 


4246i5 


7 


63 


984085 




58 


44o529 


8 


21 


559471 


35 


26 


425073 


7 


62 


984050 




58 


441022 


8 


20 


558978 
558486 


34 


2 


42353o 


7 


6 1 


984015 




58 


44i5i4 


8 


»9 


33 


423987 


7 


60 


983981 




58 


442006 


8 


l 9 


557994 


32 


29 


426443 


7 


60 


988946 




58 


442497 


8 


18 


5575o3 


3i 


3o 


426899 


7 


5 9 


983911 




58 


442988 


. 8 


'7 


557012 


3o 


3i 


9-427354 


7 


58 


9-983075 




58 


9-443479 


8 


16 


IO-55652I 


29 


32 


427809 


7 


5 7 


983840 




I 9 


443968 


8 


16 


556o32 


28 


33 


428263 


7 


56 


9838o5 




5 9 


444458 


8 


i5 


555542 


27 


34 


428717 


7 


55 


983770 




a 9 


444947 


8 


14 


555o53 


26 


35 


429170 


7 


54 


983735 




5 . 9 


445435 


8 


i3 


554565 


25 


36" 


429623 


7 


53 


983700 




5 9 


445923 


8 


12 


554077 


24 


3? 


430075 


7 


52 


983664 




5 9 


44641 1 


8 


12 


553589 


23 


38 


43o527 
430978 


7 


52 


983629 




39 


446898 


8 


ii 


553io2 


22 


3 9 


7 


5i 


983594 




39 


447384 


8 


10 


5526i6 


21 


40 


431429 


7 


5o 


983558 




-V 


447870 


8 


09 


552i3o 


20 


41 


9-431879 


7 


49 


9-983523 




•>v 


9-448356 


8 


09 


io-55i644 


'9 


42 


432329 


7 


49 


983487I 


P 


448841 


8 


08 


55i 109 


18 


43 


432778 


7 


43 


983432 




5 9 


449326 


8 


07 


550674 


17 


44 


433226 


7 


47 


983416 




5 9 


449810 


8 


06 


550190 


16 


45 


433675 


7 


46 


98338i 




5 9 


450294 


8 


06 


549706 


i5 


46 


434122 


7 


45 


983345 




39 


450777 


8 


o5 


549223 


14 


% 


434569 


7 


44 


983309 




39 


45i26o 


8 


04 


548740 


i3 


435oi6 


7 


44 


983278 




60 


45i743 


8 


o3 


548257 


12 


49 


435462 


7 


43 


983238 




60 


452225 


8 


02 


54777 5 


11 


5c 


433908 


7 


42 


983202 




60 


452706 


8 


02 


547294 


. 10 


5i 


9-436353 


7 


41 


9-983166 




60 


9-453187 


8 


01 


10 -5468 1 3 


I 


52 


436798 


7 


40 


983i3o 




60 


453668 


8 


00 


546332 


53 


437242 


7 


40 


983094 




60 


454U8 


7 


99 


545852 


1 


54 


437686 


7 


3 9 


983o58 




60 


454628 


7 


99 


545372 


6 


55 


438129 


7 


38 


988022 




60 


455107 


7 


9 8 


544898 


5 


56 


43857-2 


7 


37 


982986 




00 


455586 


7 


97 


544414 


4 


5 7 


439014 


7 


36 


982930I 


60 


456064 


7 


96 


543936, 3 


58 


439456 


7 


36 


982914 




60 


456542 


7 


96 


543458 


2 


59 


439897 


7 


35 


982878 




60 


457019 


7 


9 5 


542981 


1 


60 


44o338 
Cor* i ue 


7-34 


982842 
hi no 


60 


437496 
Cotang. 


7 


94 


542604 

"TangT 





1). 


7 


l-> 


" J 


r~ 



34 


(16 


DEGREES.) A 


TABLE OF LOGARITHMIC 




M. 


Sine 


D. 


Cosine 


I). 


Taiiff. 


D. 


Cotanor. 







9-44o338 


7-34 


9-982842 


•6o 


9-457496 


7'94 


io-54?5o4 


60 


i 


440778 


7 


33 


982805 


■ 60 


457973 


7 


93 


542027 


5 


2 


44i2 18 


7 


32 


982769 


■61 


458449 


7 


9 3 


54i55i 


3 


44i 658 


7 


3i 


982733 


•61 


458920 


7 


92 


541075 


57 


4 


442096 


7 


3i 


982696 


•6i 


459400 


7 


91 


540600 


56 


5 


442535 


7 


3o 


982660 


•61 


469875 


7 


90 


54oi25 55 


6 


442973 


7 


29 


982624 


•6i 


46o349 


7 


1 


53 9 65i 


54 


7 


443410 


7 


28 


982587 


•6i 


46o823 


7 


539177 


53 


8 


443847 


7 


27 


982551 


•61 


461297 


7 


538703 


5a 


1 9 


444284 


7 


27 


982514 


•61 


461770 


7 


88 


53823o 


5i 


!■ jo 


-444720 


7 


26 


982477 


•61 


462242 


7 


87 


537758 


5o 


1 u 


9-445i55 


7 


25 


0-982441 


•61 


9-462714 


7 


86 


10-537286 


4o 


12 


445590 


7 


24 


982404 


•61 


463 1 86 


7 


85 


536814 


48 


i3 


446o25 


7 


23 


982367 


•61 


463658 


7 


85 


536342 


47 


14 


446459 


7 


23 


982331 


•61 


464129 


7 


84 


5358 7 i 


46 


i5 


446893 


7 


22 


982294 


•61 


464599 


7 


83 


5354oi 


45 


16 


447326 


7 


21 


982267 


•61 


465069 


7 


83 


53493i 


44 


H 


447759 




20 


982220 


•62 


46553o 
466008 


7 


82 


53446i 


43 


18 


448191 


7 


20 


982183 


-62 


7 


81 


533992 
533524 


42 


19 


448623 


7 


!3 


982146 


•62 


466476 


7 


80 


4« 


20 


449054 


7 


982109 


•62 


466945 


7 


80 


533o55 


40 


21 


9-449485 


7 


17 


9-082072 


•62 


9-467413 


7 


]l 


io-532587J 3g 


22 


4499i5 


7 


16 


982035 


.62 


467880 


7 


532120 


38 


23 


45o345 


7 


16 


981998 


.62 


468347 


7 


78 


53 1 653 


ll 


24 


450775 


7 


i5 


98 1 96 1 


•62 


468814 


7 


77 


53i 186 


25 


45 1 204 


7 


14 


981924 


•62 


469280 


7 


76 


530720 


35 


26 


45i632 


7 


i3 


981886 


•62 


469746 


7 


75 


53o254 


34 


27 


452o6o 


7 


i3 


981849 


•62 


470211 


7 


7 5 


529789 


33 


28 


452488 


7 


12 


981812 


•62 


470676 


7 


74 


529324 32 


I 9 


45291 5 


7 


u 


981774 


•62 


471 Ui 


7 


73 


52885 9 3i 


3o 


453342 


7 


10 


981737 


•62 


471605 


7 


73 


5 2 83 9 5 


3o 


3i 


9-453768 


7 


10 


9-981699 


•63 


9-472068 


7 


72 


10.527932 


2 


32 


454194 


7 


09 


981662 


•63 


472532 


7 


7i 


527468 


33 


454619 


7 


08 


981625 


•63 


472995 


7 


7' 


527005 


27 


•34 


455o44 


7 


07 


9 8i58 7 


•63 


473457 


7 


70 


526543 


26 


35 


455469 


7 


07 


981549 


•63 


473919 


7 


69 


526081 


25 


36 


4558 9 3 


7 


06 


981512 


•63 


47438i 


7 


% 


5256iq 


24 


)U 


4563 16 


7 


o5 


98i474 


•63 


474842 


7 


5 2 5i58 


23 


456 7 3 9 


7 


04 


981436 


•63 


4753o3 


7 


67 


524697 


22 


I 39 


457162 


7 


04 


981399 


•63 


475763 


7 


07 


524237 


2t 


. 40 


45 7 584 


7 


o3 


981 36 1 


•63 


476223 


7 


66 


523777 


20 


41 


9-458oo6 


7 


02 


9 -98 1 323 


•63 


9-476683 


7 


65 


io-5233i7 


>9 


42 


4584-27 


7 


01 


981285 


•63 


477U2 


7 


65 


522858 


18 


43 


458848 


7 


01 


981247 


•63 


477601 


7 


64 


522399 


'7 


44 


459268 


7 


00 


981209 


•63 


478059 


7 


63 


521941 


16 


45 


459688 


6 


$ 


981 171 


•63 


478517 


7 


63 


521483 


i5 


46 


460108 


6 


98n33 


-64 


478975 


7 


62 


521025 


14 


47 


460527 


6 


98 


981095 


•64 


479432 


7 


61 


52o568 


i3 


48 


460946 


6 


97 


981057 


•64 


479889 
48o345 


7 


61 


5201 11 


12 


49 


46 1 364 


6 


96 


981019 


.64 


7 


60 


5 i 9 655 


ii 


5o 


461782 


6 


95 


980981 


.64 


480801 


7 


5 9 


519199 


10 


5i 


9-462199 


6 


9 5 


9-980942 


• 64 


9-481257 


7 


5 9 


10-518743 


§ 


02 


462616 


6 


94 


980904 


.64 


481712 


7 


58 


518288 


53 


463o32 


6 


9 3 


980866 


•64 


482167 


7 


^ 


5i7833 


7 


54 


463448 


6 


9 3 


980827 


.64 


482621 


7 


% 


517379 


6 


55 


463864 


6 


92 


980789 


-64 


483075 


7 


56 


5l6925 3 | 


56 


464279 


6 


9' 


980750 


-64 


483529 


7 


55 


» 5i647» 4 ] 


ll 


464694 


6 


90 


980712 


•64 


' 483982 


7 


55 


5i6oi8 3 ; 


465 1 08 


6 


90 


980673 


•64 


484435 


7 


54 


515565 1 a | 


5 9 


46552 2 


6 


ll 


980635 


.64 


484887 


7 


53 


5i5ii3 


1 j 


60 


465935 


6 


980596 


•64 


48533 9 


7 53 


514661 
_ Tang7~ 


* ! 


' 


Cosine 


D. 


Sine 


73° 


Cotang. 


'i>. 


Mj 





SINES AND TANGENTS 


(17 DEGREES. 


) 


35 


1 ° 


| Sine 


D. 


Cosine 


B. 


Tang. 


D. 


Cotang. 


60 


9-465q35 
466348 


6-88 


9-980596 


• 64 


9-48533<3 


7-55 


lo-5i466l 


i 


6 


• 88 


9 8o558 


• 64 


485791 


7'52 


5 1 4209 
5i3758 


59 

58 


! \ 


466761 


6 


t 


980519 


•65 


486242 


7 .5i 


1 3 


467173 


6 


980480 


• 65 


486693 


7-5i 


5i33o7 


5 7 


1 4 


46708a 


6 


•85 


980442 


•65 


487143 


7-00 


512857 


56 


1 5 


467996 


6 


•85 


980403 


•65 


487593 


7-49 


512407 


55 


i 6 


468407 


6 


• 84 


980364 


•65 


488043 


7-49 


5n 9 57 


54 


1 I 


468817 


6 


• 83 


980325 


•65 


488492 


7-48 


5n5o8 


53 


8 


469227 


6 


• 83 


980286 


•65 


488941 


7-47 


5no59 


52 


9 


469637 


6 


•82 


980247 


•65 


489390 


7-47 


5 i 06 10 


5i 


10 


470046 


6 


•81 


980208 


•65 


48 9 838 


7-46 


5ioi62 


5o 


a 


9-470455 


6 


.80 


9-980169 


• 65 


9-490286 


7.46 


10-509714 


49 


12 


470863 


6 


• 8o 


980130 


-65 


490733 


7.45 


509267 
5o882o 


48 


i3 


471271 


6 


]l 


9S0091 


• 65 


491 180 


7-44 


47 


14 


471679 


6 


980002 


-65 


491627 


7-44 


5o8373 


46 


i5 


472086 


6 


78 


980012 


• 65 


492073 


7 . 4 3 


507927 


45 


16 


472492 


6 


77 


979973 


• 65 


492519 


7-43 


5o 7 48i 


44 


\l 


472898 


6 


76 


979934 


• 66 


492965 


7-42 


5o7o35 


43 


4733o4 


6 


76 


979890 


• 66 


493410 


7-4i 


506590 


42 


l 9 


473710 


6 


75 


979805 


• 66 


493854 


7-40 


5o6i46 


41 


20 


474i 1 5 


6 


74 


979816 


• 66 


494299 
9.494743 


7 • 4° 


505701 


40 


21 


9-474019 


6 


74 


9.979776 


.66 


7-40 


io-5o5257 


S 


22 


474923 
475327 


6 


73 


979737 


.66 


490186 


7-39 
7-38 


5*04814 


38 


23 


6 


72 


979697 


.66 


49563o 


504370 


37 


24 


475730 


6 


72 


979658 


.66 


496073 


7-37 


508927 


36 


20 


476i33 


6 


7i 


979618 


.66 


4965i5 


7-37 


5o3485 


35 


26 


476536 


6 


70 


979579 


.66 


496957 


7-36 


5o3o43 


34 


2 7 


476938 


6 


69 


979539 


• 66 


497399 


7.36 


5o26oi 


33 


2a 


477340 


6 


69 


979499 


• 66 


497841 


7-35 


5o2i5o 
501718 
501278 


32 


29 


477741 


6 


68 


9794^9 


• 66 


498282 


7-34 


V 


3o 


478142 


6 


67 


979420 


• 66 


498722 


7-34 


3o 


3i 


9-478542 


6 


67 


9 -97 9 33o 


• 66 


9.499163 


7-33 


io-5oo837 


29 


32 


478942 


6 


66 


979340 


.66 


499603 


7-33 


5oo3o7 
499908 
499319 


28 


33 


479342 


6 


65 


979300 


.67 


600042 


7.32 


27 


34 


479741 


6 


65 


979260 


.67 


5oo48i 


7-31 


26 


35 


480140 


6 


64 


979220 


.67 


500920 


7-3i 


499080 
498641 


20 


36 


48o539 


6 


63 


979180 


.67 


5oi359 


7.30 


24 


37 


480937 


6 


63 


979140 


.67 


501797 


7-3o 


498203 


23 


38 


48i334 


6 


62 


979100 


.67 


5o2235 


7.29 


497765" 


22 


3 9 


481731 


6 


61 


979059 


.67 


502672 


7.28 


497328 


21 


40 


482128 


6 


61 


970019 


.67 


5o3io9 


7.28 


496891 
10^4.96404 


20 


41 


9*482525 


6 


60 


9-978979 


.67 


9'5o3546 


7.27 


'9 


42 


482921 


6 


5 9 


978939 


.67 


503982 


7.27 


496018 


18 


43 


4833 1 6 


6 


5q 


978898 


.67 


5o44i8 


7-26 


495582 


17 


44 


483712 


6 


58 


97 8858 


.67 


504804 


7-25 


495146 


16 


45 


484107 


6 


57 

il 

56 


978817 


.67 


505289 


7-25 


494711 


i5 


46 


4845oi 


6 


97^777 


.67 


5o5724 


7-24 


494276 


14 


47 


484895 


6 


978736 


il 


5o6i59 


7 • 24 


493841 


i3 


48 


485289 


6 


55 


973696 


506093 


7 . 2 3 


493407 


12 


i i 9 


485682 


6 


55 


978605 


• 68 


507027 


7-22 


492973 


1 1 


So 


486075 


6 


54 


978615 


•63 


507460 


7-^2 


492040 


10 


5i 


9-486467 


6 


53 


9-978574 


• 68 


9007893 


7-21 


10-492107 


9 


02 


486860 


6- 


53 


978533 


•68 


5o8326 


7-21 


491674 


8 


53 


487251 


6- 


52 . 


978493 


• 68 


508709 


7-20 i 


491241 


7 


54 


487643 
488o34 


6- 


5i 


978452 


• 68 


509191 


7-19 


490800 
490378 
489946 


6 


55 


6- 


5i 


97841 1 


•68 


509622 


?:3 


5 


56 


488424 


6- 


5o 


978370 


•68! 


5ioo54 


4 


tl 


488814 


6 


5o 


97 832o 


•68! 


5 10485 


7.18 


4895 1 5| ; 


489204 


6 


% 


978288 


•68! 


510916 


7-17 i 


489084 

488654 


2 


i 9 


489093 
489982 
_Casiuy 


6- 


978247 


• 68l 


5n346 


7.16 


I 


60 


6- 


48 


978206 
Sme 


■68| 


511776 


7. .6 


48S224 
Tansr. ' 


3 


'_~p._; 


7~2° 1 


Cotaiitf. 


D. 1 



36 


(18 DEGREES.) A 


TABLE OF LOGARITHMIC 




~M.' 


i Sine 


D. 


Cosine ) D. 


Tang. 


D. 


Cotaug. 


1 





| 9~- 489982 


6.48 


9-978206' .68 


9-5n 77 6 


7.16 


10-488224 


60 


» 


490371 


6-48 


978165; -68 


5 1 2206 


7- if) 


487794 


!? 


2 


490759 


6-47 


978124 -68 


5i2635 


7-i5 


487365 


58 


3 


49 11 47 


6-46 


978083 -69 


5 1 3064 


7-i4 


486 9 36 


57 


4 


49i535 


6-46 


978042 -69 


5 1 3493 


7-i4 


486507 


56 


5 


491922 


6-45 


978001 -69 


513921 


7 -i3 


486079 55 


6 


492308 


6-44 


977959 -69 


514349 


7-. 3 


48565 1 


54 


7 


492695 


6-44 


977918 

977877 


.69 


5U777 


7-12 


485223 


53 


8 


493o8i 


6-43 


.69 


616204 


7-12 


484796 


5:> 


9 


493466 


6-42 


977835 


.69 


5i563i 


7. 11 


484369 


5i 


10 


49385 1 


6-42 


977794 


.69 


5i6o5"7 


7- 10 


483 9 43 


5o 


ii 


9-494236 


6-41 


9-977762 


.69 


9-316484 


7-10 


io-4835i6 


49 


12 


49462 1 


6-41 


977711 


.69 


516910 
5i 7 335 


7-09 


483090 


48 


i3 


490005 


6-40 


977669 


•69 


7.09 


482665 


47 


U 


495388 


6-3 9 


977628, -69 


517761 


7-08 


48223a 


46 


i5 


493772 


6-3 9 
6-38 


977586 -69 


5i8i85 


7-08 


48i8i5 


45 


16 


496154 


9 77544 : -70 


5i86i 


7-07 


481390 


44 


17 


496537 


6-3 7 


9775o3 ( -70 


519034 


7-06 


480966 


43 


18 


4969 1 9 


6-3 7 


977461 j -70 


5 1 9458 


7-06 


480D42 


42 


19 


497301 


6-36 


977419, -70 


5i 9 88 2 


7-o5 


4801 18 


4i 


20 


497682 


6-36 


977377] -7° 


52o3o5 


7 -o5 


479 6 9 5 


40 


21 


9 ■ 498064 


6-35 


9-977335 -70 


9-520728 


7-o4 


10-479272 


3 9 


22 


498444 


6-34 


977293 -70 


52ii5i 


7 -o3 


478849 


38 


23 


498825 


6-34 


9772Di| -70 


52i573 


7 -o3 


478427 


37 


24 


499204 


6-33 


977209; -70 


521995 


7 -o3 


478006 


36 


2J 


499584 


6-32 


977167 -70 
977126 -70 


522417 


7-02 


477583] 35 


26 


499963 


6-32 


522838 


7-02 


477 l62 | 34 


2 7 


5oo342 


6-3i 


977083 


•70 


52325 Q 


7-01 


476741 33 


23 


500721 


6-3i 


977041 


•70 


52368o 


7-01 


476320 32 


o 9 


501099 


6-3o 


976999 -70 


524100 


7-oo 


475900 3i 


3o 


501476 


6-29 


976957 


•70 


52452o 


6.99 


475480 3o 


3i 


9-5oi854 


6-29 
6-28 


9-976914 

976812 
97683o 


•70 


9 -52493 9 


6.99 


10-475061] 29 


32 


50223l 


•71 


52535g 

525778; 


6- 98 


474641 j 28 


33 


502607 


6-23 


•71 


6- 9 3 


474222 27 


34 


502984 


6-27 


976787 


•71 


526197 


6.97 


4738o3 


26 


• 35 


5o336o 


6-26 


970745 


•71 


526615 


6-97 


473385 


25 


36 


5o3735 


6-26 


976702 


•71 


527033! 


6-96 


472967 


24 


37 


5o4no 


6-25 


976660 


•71 


52 74 5i 


6-96 


472549 


23 


38 


5o4485 


6-25 


976617 


•71 


527868' 


6-g5 


472132 


12 


3 9 


504860 


6-24 


976574 


•71 


528285' 


6- 9 5 


47171 5 


21 


4o 


5o5234 


6-23 


976532 


•71 


528702 


6-94 


471298 


20 


4i 


9-5o56oS 


6-23 


9-976489 -71 


•9-529119! 


6- 9 3 


10-470881 


19 


42 


O05981 


6-22 


976446 


•7» 


529535 


6- 9 3 


470465 


18 


43 


5o6354 


6-22 


976404 


•7 1 


52995o ! 


6- 9 3 


4700D0 


17 


44 


506727 


6-21 


976361 


■7i 


53o366] 


6-92 


469634 


16 


45 


607099 


6-20 


676318 -71 


530781! 


6-91 


469219 


i5 


46 


507471 


6-20 


976275J -71 


53i 196 


6-91 


468804 


14 


47 


607843 


6- 19 


976232; -72 


53 161 1 i 


6-90 


468389 


i3 


48 


5o82i4 


6- 19 
6- 18 


976189' .72 


532025 


6-90 


467975 


12 


| *9 


5o8585 


976i46 [ .72 


53243g 


6-89 


467561 


11 


I DO 


608966 
9009326 


6-i8 


976103. -72 


532853 


6-89 


467147 


10 


5i 


6-17 


9-976060; -72 


9-533266 


6-88 


10-466734 


i 


52 


509696 


6-i6 


976017; -72 


533679 


6-88 


466321 


53 


5joo65 


6- 16 


9739741 .72 


534092 


6-87 


466908 


7 


54 


610434 


6-i5 


975930 -72 


5345o4 


6-87 


465496 
465o84 


6 


55 J 


5io8o3 


6-i5 


975887 .72 


534916 


6-86 


5 


56 


511172; 


6-14 


975844 -72 


535328 


6-86 


464672 


4 


i ? 7 


5n54o 


6- 13 


9758oo ; -72 


535739 


6-85 


464261 


3 


? 8 


511907 


6- 13 


975767 -72 


536 1 5o 


6-85 


46385o 2 


*9 | 


512275; 


612 


975714 -72 


53656i i 


6.S4 ; 


463439 1 
463028; 1 


fro i 


5 1 2642 | 


6-12 


975670 -72 


536972 


6-84 1 




Cosine 1 


___„_... 


Sine 


• 1 


Cotaug. i 


D. 


Tang. ' M. j 



BINES AND TANGENTS. (19 DEGREES.) 



37 



r iL 


| Sine 


1 D ' 


Cosine 


•73 


Tang. 


1 D- 


Cotang. | 


o 


9-612642 


| 6-12 


9-975670 


9-536972 


j 6-84 


10-463028 


60 


I 


5i3oog 


1 6-H 


975627 


-73 


537382 


1 6-83 


462618 


5 9 


2 


5i3375 


1 6-n 


975583 


•73 


537792 


6-83 


462208 


58 


3 


5i374i 


6-io 


975539 


•73 


538202 


6-82 


461798 


57 


4 


614107 


6-09 


97549 6 


-73 


5386ii 


6-82 


461389 


56 


5 


5i4472 


^•09 


975452 


•73 


539020 


6-8x 


460980 
46o5 7 i 


55 


6 


5U837 


6-o8 


975408 


•73 


539429 


6-81 


54 


7 


5l3202 


6-o8 


975365 


•73 


53 9 837 


6-8o 


46oi63 


53 


8 


5i5566 


6-07 


975321 


•73 


540245 


6-8o 


459755 


52 


9 


5i5g3o 


6-07 


976277 


•73 


54o653 


6-79 


45o347 


5! 


IO 


616294 


6-o6 


975233 


•73 


641 06 1 


6-79 


458939 


5o 


u 


9' 516607 


6-o5 


9-9^189 


■73 


9-541468 


6-78 


IO-458532 


49 


12 


517020 


6-o5 


975145 


•73 


54i8 7 5 


6-78 


458125 


48 


i3 


517382 


6-o4 


975101 


•73 


542281 


6-77 


457719 


47 


U 


5i7745 


6-o4 


976057 


•73 


542688 


6-77 


467312 


46 


i5 


518107 


6-o3 


975oi3 


•73 


543094 


6-76 


466906 
4565oi 


45 


16 


518468 


6-0.3 


974969 


•74 


643499 


6-76 


44 


17 


518829 


6-o2 


974925 


■74 


543905 


6- 7 5 


456095 


43 


18 


519190 


6-oi 


974880 


•74 


5443 10 


' 6- 7 5 


466690 
455285 


42 


*9 


5i955i 


6-oi 


974836 


•74 


5447i5 


6-74 


4i 


20 


5 1 99 1 1 


6-oo 


97479 2 


•74 


546119 


6-74 


454881 


4o 


21 


9-520271 


6-oo 


9-974748 


•74 


9-545524 


6-73 


10-454476 


ll 


22 


52o63i 


5-99 


974703 


•74 


546928 


6- 7 3 


454072 


23 


620990 


5.99 


974669 


•74 


54633 1 


6-72 


45366 9 
453265 


37 


24 


52i349 


5.98 


974614 


•74 


546735 


6-72 


36 


25 


521707 


5.98 


974670 


•74 


547i38 


6-71 


462862 


35 


26 


522066 


5-97 


974625 


•74 


54754o 


6-71 


462460 


34 


27 


522424 


5- 9 6 


97448i 


•74 


547943 


6-70 


452o57 


33 


28 


522781 


5-96 


974436 


•74 


, 548345 


6-70 


45 1 65 5 


32 


29 


• 523i38 


5-9D 


974391 


•74 


548747 


6-69 


45i253 


3i 


3o 


523495 


5- 9 5 


974347 


• 7 5 


549 1 49 


6-69 


460861 


3o 


3i 


9-523852 


6-94 


9-974302 


•75 


9-549550 


6-68 


10-450460 


2 


32 


524208 


5-94 


974267 


• 7 5 


549951 
55o352 


6-68 


460049 


33 


524564 


5- 9 3 


974212 


•75 


6-67 


449648 


27 


34 


524920 


5- 9 3 


974i67 


• 7 5 


55o752 


6-67 


449248 
448848 


26 


35 


525275 


6-92 


974122 


.75 


55ii52 


6-66 


25 


36 


52563o 


5-91 


974077 


•75 


55i552 


6-66 


448448 


24 


37 


525984 


5-91 


974032 


■ 75 


551952 
55235i 


6-65 


448048 


23 


33 


526339 


5-90 


973987 


•75 


6-65 


447649 


22 


3 9 


526693 


6-90 
5-8g 


973942 


.75 


562750 


6-65 


44725o 


21 


4o 


527046 


973897 


• 75 


553i49 


6-64 


44685 1 


20 


4i 


9-527400 


5-8 9 


9-973S52 


•75 


9.553548 


6-64 


IO-446452 


19 


42 


627753 


5-88 


973807 


•73 


553946 


,6-63 


446o54 


18 


43 


528io5 


' 5-88 


973761 


.75 


554344 


6-63 


445656 


17 


44 


528458 


5-87 


973716 


.76 


554741 


6-62 


446259 


16 


45 


5288io 


5-8 7 


973671 


•76 


555i3 9 


6-62 


444861 


i5 


46 


529161 


5-86 


973625 


•76 


555536 


6-61 


444464 


14 


% 


5295i3 


5-86 


97 358o 


•76 


555q33 


6-6i 


444067 


i3 


629864 


5-85 


973535 


•76 


55632Q 
5567 2 5 


6-6o 


4436 7 i 


12 


i 9 


53o2i5 


5-85 


973489 


.76 


6-6o 


443275 


n 


5o 


53o565 


5-84 


.973444 


•76 


567121 


6-59 


442879 
10-442483 


10 


5i 


9-53o9i5 


5-84 


9-973398 


•76 


9-567517 


6- 5 9 


I 


52 


53i265 


5-83 


973352 


•76 


557913 


6-5o 
6-58 


442087 


53 


53i6i4 


5-82 


973307 


•76 


5583o8 


441692 


1 


54 


53i 9 63 


5-82 


973261 


•76 


558702 


6-58 


441298! 6 


55 


5323 1 2 


5-8! 


9732i5 


•76 


559097 


6-57 


440903 1 5 


56 


53266i 


5-8i 


9731691 


.76 


569491 


6- 5 7 


44o5oo 4 
4401 1 5| 3 


5i 


533009 


5-8o 


973124 


.76 


55 9 885 


6 • 56 


58 


533357 


5. So 


973078, 


•76 


560279 


6-56 


439721 


2 


5 9 


533704 


i.% 


973082) 


•77 


560673 


6-55 


43 9 327 


1 


6o 


534o52 


972986J 


•77 


56 1 066 


6-55 


438 9 34 





Cosine 


]). 


Sine |70° 


Cotang. 


D. 


Tung." 



38 


(20 


DEGREES.) A 


TABLE OF LOGARITE1MIC 


1 ' 


M. 


Sine 


D. 


Cosine 


LA 

! -77 


1 Tang. 
J 9.561066 


1>. 


1 Cotang. 





9«534o52 


5- 7 8 


9-972986 


' 6-55 


10-438934! 60 


i 


534399 


5-77 


972940; -77 


56i45g 


6-54 


438341J 59 
438i49 58 


2 


334745 


2"" 


972894 -77 


56 1 85 1 


6-54 


3 


535092 


I'V 


972848 


•77 


562244 


6-53 


437756- 5 7 


4 


53543S 


5.76 


972802 


•77 


562636 


6-53 


437364) 56 


5 


535783 


5.76 


972755 


•77 


563028 


6-53 


436972) 55 
43658i 54 


6 


536129 


5- 7 5 


972709 


•77 


563419 


6-52 


I 


536474 


5-74 


972663 -77; 5638n 


6-52 


436i8q| 53 
433798 52 


5368 1 8 


5.74 


972617 


•77 


564202 


6-5i 


9 


537i63 


5.73 


972570 


•77 


564392 


6-5i 


433408 


5i 


10 


537507 


5- 7 3 


972524 


•77 


564 9 83 


6-5o 


435oi7 


5c 


u 


9-53785i 


5-72 


9.972478 


•77 


9-565373 


6-5o 


10-4346^1 


49 


12 


538 194 
538538 


5-72 


97243i 


•78! ■ 565 7 63 


6-49 


434237 48 


i3 


5.71 


972385 


.78 


566 1 53 


6-49 


433847 47 


14 


53888o 


5.71 


972338 


•78 


566542 


6-49 


433458 


46 


i5 


539223 


5-70 


972291 


.78 


566932 
567320 


6-48 


433o68 


45 


16 


53g565 


5-70 


972245 


•78 


6.48 


432680 


44 


i'7 


539907 


5-69 . 


972198 
972i5i 


"% 


567709 
568098 
568486 


6-47 


432291 


43 


i3 


540249 


5-6 9 
5-68 


' 7 S 


6-47 


43 1 902 j 42 


19 


540590 


972105 


.78 


6.46 


43i3i4 4f 


20 


54093 1 


5-68 


972o58 


.78 


5688 7 3 


6-46 


43 1 1 27 40 


21 


9-541272 


5-67 


9-972011 


.78 


9.569261 


6-45 


10-430739: 3a 
43o352i 38 


22 


54i6i3 


5-67 


971964 


.78 


569648 


6-45 


23 


541953 


5-66 


971917 
971870 


■ l8 a 


570035 


6-45 


429965J 37 


24 


542293 
542632 


5-66 


.78 


570422 


6-44 


429378 36 


25 


5-65 


971823 


.78 


570809 
571193 


6-44 


429191; 35 


.26 


542971 


5-65 


971776 


•78 


6-43 


4288o5, 34 


27 


5433io 


5-64 


971729 


•79 


571681 


6-43 


428419 s 33 


28 


543649 


5-64 


971682 


•79 


571967 
572352 


6-42 


428o33) 32 


29 


543987 
544325 


5-63 


971635 


•79 


6-42 


427648 3 1 


3o 


5-63 ' 


971588 


•79 


572738 


6-42 


427262 3o 


?* 


q • 544663 


5-62 


9-971540 


•79 


9.573123 


6-41 


10-426877) 29 


32 


545ooo 


5-62 


971493 


•79 


573507 


6-4i 


426493 28 


33 


545338 


5-6i 


971446 


•79 


573892 


6-40 


426108; 27 


34 


5456 7 4 


> 61 


971398 
97!35i 


•79 


574276 


6-40 


425724 26 


35 


54601 1 


5 60 


•79 


574660 


6.39 


42534o : 25 


36 


546347 


5.6o 


97i3o3 


•79 


575044 


6-3 9 


4249661 24 

424373 23 


37 


546683 


5-5 9 


971256 


•79 


675427 


6-3 9 


38 


547019 


5-5 9 
5-58 


971208 


•79 


573810 


6-33 


424190! 22 


3 9 


547354 


971161 


•79 


576193 


6-33 


423807: 21 


4o 


547689 


5-58 


97iu3 


•79 


576576 


6-37 


423424! 20 


4i 


9-548024 


5.57 


9-971066 


•80 


9-576968 


6-37 


io-423o4i 19 
42265 9 18 


42 


54835 9 ' 


5-5 7 


971018 


.80 


577341 


6-36 


43 


548693, 


5-56 


970970 


■80 


5 777 23 


6-36 


422277; 17 


44 


549027 


5-56 


970922 


-8o 


578104 


6-36 


421896I 16 


45 


54g36o 


5-55 


970874 -8o 


578486 


6-^5 


42i5i4 ! i5 


46 


549693 


5-55 


970827 -8o 


578867! 


6-35 


421 i33i 14 


47 


55oo26 


5-54 


970779, -8o 


579248 


6-34 


420762 i3 


48 


55o35g 


5-54 


97073 1 -8o 


579629 


6-34 


420371 


12 


iq 


550692 


5-53 


970683 -8o 


580009! 


6-34 


419991 


11 


DO 


55io24 


5-53 


970635' -8o; 


58o38 9 


6-33 


41961 1 


10 


5i 


9-55i356' 


5-52 


9-970586] -8o; 


9-580769! 


6-33 


10-419231 

4i885i 


I 


32 


55i68 7 ! 


5-52 


970538, -8o 


58ii49 ! 


6-32 


53 


552018 


5-52 


970490I -8o 


58 1 528 


6-32 


418472 


1 


54 


f5 2 3_, 9 


5-5i 


970442 -8o, 


)8 1 907.! 


6-32 


418093 


6 


55 


55268o 


5-5i 


970394 


• 8o 


^82286! 


6-3i 


4i77i4 


5 


56 


553oio 


5-5o 


970345 


-8i 


582663 


6-3i 


417333 4 J 


5 7 1 


553341 


5-5o 


970297, 


• 81 


583043 


6-3o 


416957 3 


58 1 


553670 


5-49 


^70249 


.81 


583422 


6-3o 


4i65 7 8 2 j 


5y| 


554ooo 


5- 49 


970200 


• 8i 


5838oo ' 


6-29 


416200 1 1 


6o | 


554329 
CoRiue 1 


5-48 


970152 


• 8i 


584i77| 


6-29 

r~dl 


415823! ) 1 
Tang. jli.J 


1 


D. 


.Sine |6»Q| 


Cotang. ( 





SINES 


AND TANGENTS. 


(21 DEGREES.] 


► 


313 





Sine 


D. 


Cosiue 


D. 


Tang. | D. 


Cotang. 




9-554320 

554658 


5 


•48 


9-970152 


.81 


9084177 6.29 


io-4i5823 


"oo~ 


i 


5 


•48 


970103 


.81 


584553 6 


2 


4i5445 


u 


2 


5549»7 
5553i 3 


5 


47 


970033 


.81 


584932! 6 


4i5o68 


3 


5 


47 


970006 


.81 


585309! 6 


28 


414691 


57 


4 


555643 


5 


46 


969937 


.81 


585686' 6 


27 


4i43i4 


56 


5 


555971 


5 


46 


969909 


.81 


586062 


6 


27 


4i3 9 38 


55 


6 


556299 


5 


45 


969860 


• 8i 


58643o 
5868i5 


6 


27 


4l356i 


54 


7 


556626 


' 5 


45 


96981 1 


.81 


6 


26 


4i3i85 


53 


8 


556g53 


5 


44 


969762 


.81 


587190 


6 


26 


412810 


52 


9 


557280 


5 


44 


969714 


.81 


587566: 6 


25 


412434 


5i 


10 


557606 


5 


43 


969665 


• 81 


587941 ! 6 


25 


412059 


5o 


ii 


9-557932 


5 


43 


9-969616 


.82 


9-5883i6 ; 6 


25 


10-411684 


49 


12 


558258 


5 


43 


969567 


.82 


5886 9 i 


6 


24 


411309 


48 


i3 


558583 


5 


42 


969518 


.82 


589066 


6 


24 


410934 


47 


14 


558909 


5 


42 


969469 


• 82 


589440 


6 


23 


4io56o 


46 


i5 


559234 


5 


41 


969420 


.82 


589814 


6 


23 


410186 


45 


16 


55 9 558 


5 


4i 


969370 


.82 


590188 


6 


23 


409812 


44 


iZ 


55 9 883 


5 


40 


969321 


.82 


590562 


6 


22 


409438 


43 


560207 


5 


40 


969272 


-82 


590935 


6 


22 


409065 


42 


'9 


56o53i 


5 


3o 


969223 


.82 


591308 


6 


22 


408692 


4i 


20 


56o855 


5 


3 9 


969173 


.82 


591681 


6 


21 


4o83ig 


40 


21 


9-56ii78 


5 


38 


9-969124 


.82 


9-592054 


6 


21 


10-407946 
407374 


£ 


22 


56i5oi 


5 


38 


969073 


.82 


592426 


6 


20 


23 


561824 


5 


5 7 


969025 


.82 


592798 


6 


20 


407202 


37 


24 


562146 


5 


37 


968976 


•82 


593170 


6 


'9 


406829 


36 


25 


562468 


5 


36 


968926 


•83 


593542 


6 


19 


4o6458 


35 


26 


562790 


5 


36 


968877 


•83 


593914 


6 


18 


406086 


341 


2 


563ii2 


5 


36 


968827 


•83 


594285 


6 


18 


4057 1 5 


33 


563433 


5 


35 


968777 


•83 


5 9 4656 


6 


18 


4o5344 


32 


29 


563755 


5 


35 


968728 


•83 


595027 


6 


17 


404973 


3i 


3o 


564075 


5 


34 


968678 


•83 


593398 


6 


n 


404602 


3o 


3i 


9-564396 


5 


34 


9-968628 


•83 


9 -595768 


6 


17 


10-404232 


2 


32 


564716 


5 


33 


96SD78 


•83 


5 9 6i38 


6 


16 


4o3862 


33 


565o36 


5 


33 


9 6S52S 


•83 


5g65o8 


6 


16 


■ 403492 


27 


34 


565356 


5 


32 


968479 


•83 


596878 


6 


16 


4o3i22 


26 


35 


5656 7 6 


5 


32 


968429 


•83 


597247 


6 


i5 


402753 


25 


36 


565995 
5663i4 


5 


3i 


968379 


•83 


597616 


6 


i5 


402384 


24 


& 


5 


3i 


96S329 


•83 


597985 


6 


i5 


40201 5 


23 


566632 


5 


3i 


968278 


•83 


5o8354 


6 


14 


401646 


22 


3 9 


566 9 5i 


5 


3o 


968228 


•84 


598722 


6 


14 


401278 


21 


4o 


567269 


5 


3o 


968178 


•84 


599091 


6 


i3 


400909 


20 


4i 


9- 56 7 587 


5 


29 


9-968128 


.84 


9-599459 


6 


i3 


10 -400341 


IS 


42 


567904 

568222 


5 


2 2 


968078 


.84 


599827 


6 


i3 


400173 


43 


5 


28 


968027 


.84 


600194 


6 


12 


399806 


17. 


44 


56853 9 


5 


28 


967977 


• 84 


6oo562 


6 


12 


399438 


16 


45 


568856 


5 


28 


967927 


• 84 


600929 


6 


11 


399071 


i5 


d6 


569172 


5 


27 


967876 


• 84 


601296 


6 


11 


398704 


14 


*7 


56 9 488 


5 


27 


967826 


• 84 


601662 


6 


11 


3 9 8338 


i3 


48 


569804 


5 


26 


967775 


• 84 


602029 


6 


10 


397971 12 


49 


570120 


5 


26 


967725 


• 84 


602395 


6 


10 


397605I 11 


5o 


570435 


5 


25 


967674 


• 84 


602761 


6 


10 


397239I 10 


5i 


9-570751 


5 


25 


9-967624 


.84 


9-6o3i27 


6 


09 


10-396873' 9 


52 


571066 


5 


24 


967573 


• 84 


603493 


6 


09 


396307 




53 


57i38o 


5 


24 


967522 


.85 


6o3858 


6 


09 


396142 


7 


54 


571695 


5 


23 


967471 


• 85 


604223 6 


08 


395777 


6 


55 


572009 


5 


23 


967421 


•85 


6o4588 6 


08 


393412 


5 


56 


572323 


5 


23 


967370 


•85 


604953 
6o53i7 


6 


07 


395047 


4 


5 1 
58 


5 7 2636 


5 


22 


967319 


•85 


6 


°7 


3 9 4683 


3 


572o5o 


5 


22 


967268 


•85 


6o5682 


6 


07 


3 9 43i8 


a 


59 


5 7 3 2 63 


5- 


21 


967217 


•85 


606046 


6- 


06 


393954 


1 


00 


5?35 7 5 


5-21 


967166 


•85 


606410 


6- 


06 


393590J 


Cosine 


D. 


Sine ( 


*8° 


Cotang. 


D7~ 


^arJffT 1 M. 



40 


(22 


DEGREES.) A TABLE OF lOGARITTIMrC 




M. 


tSine 


1 D ' 


Cosine | D. 


1 Tang. 


1.. 

6-o6 


1 Cotang. | 

[10.393590! 60 ! 


o 


9-573575 


1 5-21 


9-967 1661 -85 


' 9-606410 


i 


573888 


5 


• 20 


9671 i5j 


• 85 


606773 


6-06 


393227 1 5o 
392863! 5§ 


2 


574200 


5 


• 20 


967064! 


• 85 


6071.37 


6-o5 


3 


574512 


5 


• 19 


967013 


• 85 


607500 


6-o5 


392500 


57 


4 


574824 


5 


• 19 


966961 


• 85 


607863 


6-04 


392137 


56 


5 


575i36 


5 


•19 


966910 


• 85 


608225 


6-04 


391775 


55 


6 


575447 


5 


• 18 


9VJ6839 


• 85 


6o8588 


6-04 


391412 


54 


I 


575738 


5 


•18 


966808 


.85 


608930 


6-o3 


3gio5o 


53 


576069 


5 


•17 


966756 


• 86 


609J 1 2 


6-o3 


390688 


52 


9 


676379 


5 


•17 


966703 


-86 


609674 


6-o3 


390326 


5i 


10 


576689 


5 


•16 


966653 


• 86 


6,ioo36 


6-02 


389964 


5o 


ii 


9-576999 


5 


16 


9-966602 


• 86 


9.610397 


6-02 


10.389603 


40 


12 


577309 


5 


•16 


96655o 


• 86 


610739 


6-02 


3S9241 


48 


i3 


577618 


5 


• i5 


966499 


• 86 


611120 


6-oi 


38888o 


47 


14 


577927 


5 


i5 


966447 


• 86 


61 1480 


6-oi 


388520 


46 


ID 


578236 


5 


• 14 


966393 


.86 


611841 


6-oi 


388 1 5 9 


45 


16 


578545 


5 


14 


966344 


.86 


612201 


6-oo 


387799 


44 


n 


578853 


5 


■i3 


966292 


• 86 


612361 


6-oo 


38 7 43 9 


43 


18 


579162 


5 


•i3 


966240! 


.86 


612921 


6-oo 


387079 


42 


*9 


579470 


5 


•i3 


966188 


.86 


6i328i 


5-99 


386719 


41 


20 


579777 


5 


• 12 


966 1 36 


.86 


6i364i 


5.99 


38635 9 


40 


21 


9 # 58oo85 


5 


• 12 


9-966085 


.87 


9-614000 


5- 9 8 


10.386000 


3 9 


22 


580392 


- 


•11 


966033 


.87 


614339 


5- 9 8 


385641 


33 


23 


580699 
58ioo5 


5 


11 


960981 


.87 


6 1 47 1 8 


5- 9 8 


385282 


37 


24 


5 


11 


963928 


.87 


615077 


5-97 


384923 


36 


25 


58i3i2 


5 


10 


965876 


.87 


6 1 5435 


5-97 


384565 


35 


,26 


58i6i8 


5 


10 


965824 


.87 


615793 


5.97 


384207 


34 


11 


581924 


5 


09 


965772 


.87 


6i6i5i 


5- 9 6 


383849 


33 


582229 


5 


09 


965720 


.87 


616509 


5-96 


383491 
383 1 33 


32 


2 J 


58253d 


5 


09 


9 65668 


.87 


616867 


5.96 


3i 


3o 


582840 


5 


08 . 


9656 1 5 


.87 


617224 


5.93 


3S2776 


3o 


3i 


9'583i45 


5 


08 


9 -965563 


.87 


9-617582 


5.93 


10-382418 


3 


32 


583449 


5 


07 


9&55i 1 


.87 


617939 


5- 9 5 


382061 


33 


583704 


5 


07 


9 65458 


.87 


618293 


5.94 


38i 7 o5 


27 


34 


584o58 


5 


06 


965406 


.87 


6i8652 


5.94 


38 1 348 


26 


35 


58436i 


5 


06 


965353 


.88 


619008 


5.94 


380992 


2D 


36 


584665 


5 


06 


9653oi 


.88 


619364 


5. 9 3 


38o636 


24 


37 


584968 


5 


o5 


965248 


.88 


619721 


5- 9 3 


380279 


23 


38 


585272 


5 


o5 


963195 


.88 


620076 


5. 9 3 


379924 


22 


3 9 


5S55 7 4 


5 


04 


9&5i43 


.88 


6204J2 


5.92 


379D68 


21 


40 


5858 77 


5 


04 


965090 
9-965037 


.88 


620787 


5-92 


379213 


20 


41 


9-586179 


5 


o3 


.88 


9-621142 


5-92 


10-378858 


*9 


42 


586432 


5 


o3 


964984 


.88 


621497 


5-91 


3785o3 


l8 


43 


586 7 83 


5 


o3 


96493 1 


.88 


621832 


5-91 


378148 


17 


44 


587085 


5 


02 


964879 


.88 


622207 


5.90 


377793 
377439 


16 


45 


58 7 386 


5 


02 


964826 


.88 


62256i 


5.90 


i5 


46 


58 7 688 


5 


01 


964773 


.88 


622915 


5.90 


377085 


14 


% 


587989 


5 


01 


964719 


.88 


623269 


5-§ 9 


376731 


i3 


588289 


5 


01 


964666 


.89 


623623 


g-g9 


376377 


12 


49 


588590 


5 


00 


964613 


.89 


623976 


5.89 


376024 


11 


5o 


588890 


5 


00 


964560 


.89 


62433o 


5-88 


375670 


10 


5i 


9-589190 


4 


99 


9-964507 


.89 


9-624683 


5-88 


10.375317 


I 


52 


589489 


4 


99 


964454 


.89 


625o36 


5-88 


374964 


53 


589789 


4 


$ 


964400 


.89 


625388 


5.87 


374612 


7 


54 


590088 


4 


964347 


.89 


625741 


5.87 


374259 


6 


55 


590387 


4 


98 


964294 


.89 


626093 


5.87 


373907 


5 


56 


5 9 o686 


4 


97 


964240 


•8q 


626445 


5-86 


373555 


4 


u 


590984 


4 


97 


964187 


.89 


626797 


5-86 


373203 


3 


591282 


4 


97 


964133 


•89 


627149 


5-86 


372851 


2 


59 


591580 


4 


96 


964080 


-89 


627001 


5-85 | 


372490; 
372148! 


1 


60 


591878 


4- 


96 


964026 


4 


627852I 


5-85 







Cosine 


D. 1 


Sino i 


3T° 


Cotang. 1 


D. 


Tang^J M. 





SINES AND TANGENTS. 


(23 DEGREES. 


) 


41 


TKT 




Sine 


D. 


Cosine | D. 


Tang. 


D. 


1 Cotang. 




9.591878 


4-96 


9-964026 1 .85 


9-627852 


5-85 


10-372148 


60 


I 


592176 


4-9 5 


963972 


.& 


62820^ 


5-85 


371797 


5o 

58 


2 


592473 


4-95 


963919 


•89 


628554 


5-85 


371446 


3 


592770 


4- 9 5 


963b65 


- 9 c 


628905 


5-84 


371093 


57 


4 


693067 


4.94 


9638 u 


- 9 c 


629255 


5-84 


370745 


56 


5 


5 9 3363 


4.94 


9 63 7 5 7 


■ 9 c 


629606 


5-83 


370394 


55 


6 


593659 


4- Q 3 


963704 


- 9 c 


629956 


5-83 


370044 


54 


7 


593955 


4- 9 3 


96365o 


•9 C 


63o3o6 


5-83 


369694 


53 


8 


594251 


4- 9 3 


963596 


• 9 c 


63o656 


5-83 


369344 


52 


9 


594547 


4-92 


963542 


• 9 c 


63ioo5 


5-82 


368995 


5i 


IO 


594842 


4-92 


963488 


•9 C 


63i355 


5-82 


368645 


DO 


u 


9.59D137 


4-91 


9-963434 


•9 C 


9-631704 


5-82 


10-368296 


49 


12 


5 9 5432 


4-91 


963379 


.90 


632o53 


5- 81 


367947 


48 


i3 


5 9 5 7 27 


4-91 


963320 


•9° 


6324oi 


5-8i 


367D99 


47 


U 


596021 


4-90 


963271 


•90 


632730 


5- 81 


3672D0 


46 


i5 


5963i5 


4.90 


963217 


.90 


633098 


5-8o 


366902 


45 


16 


596609 
59690J 


4-8 9 


963 1 63 


.90 


633447 


5-8o 


366o53 


44 


n 


4.89 


963108 


.91 


633795 


5- 80 


366205 


43 


18 


597196 


4.89 


963o54 


.91 


634143 


5.79 


36585 7 


42 


l 9 


597490 


4-88 


962999 


•9.1 


634490 


5.79 


3655io 


41 


20 


597783 


4-88 


962945 


•91 


634838 


5.79 


365i62 


40 


21 


Q. 598075 


4-8 7 


9-962890 


.91 


9-635i85 


5.78 


10-064815 


ll 


22 


5 9 S368 


4-87 


962836 


•91 


635532 


5.78 


364468 


23 


5 9 866o 


4-87 


962781 


.91 


635879 


5.78 


364121 


37 


24 


598952 


4-86 


962727 


•91 


636226 


5.77 


363 77 4 


36 


25 


599244 


4-86 


962672 


•91 


636572 


5.77 


363428 


35 


26 


599536 


4-85 


962617 


.91 


636919 


5-77 


363o8i 


34 


2 


599827 


4-85 


962562 


.91 


637265 


5-77 


362 7 35 


33 


6001 18 


4-85 


962008 


•91 


63 7 6h 


5-76 


36238 9 


32 


29 


600409 


4.84 


962453 


•91 


637956 


5- 7 6 


362044 


3i 


3o 


600700 


4-84 


962398 


•92 


638302 


5.76 


36i6 9 8 


3o 


3i 


9.600990 


4-84 


9-962343 


•92 


. 9-638647 


5.75 


io-36i353 


29 


32 


601280 


4-83 


962288 


•92 


638992 
639337 


5.75 


36 1 008 


28 


33 


601570 


4-83 


. 962233 


•92 


5. 7 5 


36o663 


27 


34 


601860 


4-82 


962178 


•92 


639682 


5.74 


36o3i8 


26 


35 


6021 5o 


4-82 


962123 


•92 


640027 


5-74 


359973 


25 


36 


602439 


4-82 


962067 


•92 


640371 


5-74 


359629 


24 


37 


602728 


4-8i 


962012 


•92 


640716 


5- 7 3 


359284 


23 


38 


6o3oi7 


4-8i 


961957 


•92 


641060 


5- 7 3 


358940 


22 


3 9 


6o33o5 


4-8i 


961902 


•92 


641404 


5- 7 3 


3585 9 6 


21 


40 


603594 


4- 80 


961846 


•92 


641747 


5-72 


358253 


20 


4i 


9 .6o3882 


4.80 


9-961791 


•92 


9-642091 


5-72 


10-357909 


!8 


4r 


604170 


4-79 


961735 


•92 


642434 


5-72 


357566 


43 


604457 


4-79 


961680 


•92 


642777 


5-72 


357223 


n 


44 


604745 


4-79 


961624 


• 9 3 


643120 


5- 7 i 


35688o 


16 


45 


6o5o32 


4-78 


961^69 


• 9 3 


643463 


5.71 


356537 


i5 


46 


6o53i9 


4-78 


96i5i3 


•93 


6438o6 


5- 7 i 


356194 


14 


47 


6o56o6 


4-78 


961458 


•93 


644148 


5-70 


355852 


i3 


48 


605892 


4-77 


961402 


•93 


644490 
644832 


5-70 


3555io 


12 


49 


606179 


4-77 


961346 


•93 


5-70 


355i68 


11 


5o 


606465 


4-76 


961290 


•93 


645174 


5-69 


354826 


10 


5i 


9-606751 


4-76 


9-961235 


•93 


9-6455i6 


5-69 


io-354484 


I 


52 


607036 


4-76 


961 179 


•93 


645857 


5-69 


354U3 


53 


607322 


4- 7 5 


961 1 23 


•93 


646 1 99 


5-69 


3538oi 


7 


54 


607607 


4- 7 5 


961067 


•93 


646540 


5-68 


35346o 


6 


55 


607892 


4-74 


961011 


•93 


646881 


5-68 


353ii9 


5 


56 


608177 


4-74 


960955 


•93 


647222 


5-68 


352778 


i 


57 


608461 


4-74 


960899 


• 9 3 


647562 


5.67 


352438 


3 


58 


608745 


4- 7 3 


960843 


•94 


647903 


5-67 


3520^7 


2 


59 


609029 


4- 7 3 


960786! -94 


648243 


5-67 


351707 


1 


60 


609313 


4-73 


960730I -94 


648583 


5-66 


35i4i7 





L_. ... 


Cosine 


D. 


Sine lG6 c 


Cotang. 


D. 


Tang. 


m7 



42 


(24 


DEGREES.) A 


TABLE OF LOGARITHMIC 







Sine 
9-6093T3 


1 I> " 


Cosine 


D. 


Tang. 


1 D ' 


Cotang. 
10-351417 


60 


1 4-73 


9-960730 


.94 


9-648583 


5-66* 


i 


609597 


1 4-72 


960674 


.94 


648923I 5 


•66 


351077 


a 


2 


609880 


1 4-72 


960618 


.94 


649263 


5 


• 66 


350737 


3 


610164 


4-72 


960561 


.94 


649602 


5 


• 66 


300398 


57 


4 


610447 


4-71 


96o5o5 


.94 


649942 


5 


• 65 


35oo58 


56 


5 


610729 


4-71 


960448 


•94 


65o28i 


5 


• 65 


349719 


55 


6 


611012 


4-70 


960392 


•94 


65o62o 


5 


-65 


34938o 


54 


7 


61 1294 


4-70 


96o335 


.94 


650959 


5 


•64 


349041 
348703 


53 


8 


611576 


4-70 


960279 


•94 


601297 


5 


■64 


52 


9 


6u858 


4-6 9 


960222 


.94 


65 1 636 


5 


•64 


348364 


5i 


10 


612140 


4.69 


960 1 65 


•94 


651974 


5 


•63 


348026 


5o 


ii 


9-612421 


4.69 


9-960109 


- 9 5 


9'6523i2 


5 


•63 


10-347688 


% 


12 


612702 


4-68 


960052 


.95 


65265o 


5 


•63 


34"?35o 


i3 


612983 


4-68 


939995 


- 9 5 


652988 
653326 


5 


■63 


347012 


47 


U 


613264 


4-67 


93993s 


• 9 5 


5 


•62 


346674 


46 


i5 


6i3545 


4-67 


959882 


- 9 5 


653663 


5 


•C-2 


34633 7 


45 


16 


6i3825 


4-67 


959825 


- 9 5 


654ooo 


5 


•62 


346000 


44 


n 


6i4io5 


4-66 


959768 


.95 


654337 


5 


6l 


345663 


43 


18 


6U385 


4-66 


95971 1 


• 9 5 


654674 


5 


•6l 


345326 


42 


19 


6i4665 


4-66 


959654 


• 9 5 


655ou 


5 


•6l 


344989 


4i 


20 


614944 


4-65 


959596 


•9 5 


655348 


5 


•6l 


344652 


40 


21 


9-6iJ223 


4-65 


9-959539 


.?5 


9*655684 


5 


60 


ic-3443i6 


ll 


22 


6i?5o2 


4-65 


959482 


• 9 5 


656o2o 


5 


6a 


343980 


23 


615781 


4-64 


959425 


.95 


656356 


5 


6a 


343644 


37 


24 


616060 


4-64 


959368 


- 9 5 


656692 


5 


5 9 


3433o8 


36 


25 


6i6338 


4-64 


959310 


.96 


657028 


5 


5 9 


342972 


35 


26 


616616 


4-63 


959253 


.96 


657364 


a 


5 9 


342636 


34 


27 


616894 


4-63 


9 5 9 i 9 5 
959138 


.96 


657699 
658o34 


5 


5 9 


3423oi 


33 


28 


617172 


4-62 


.96 


5 


58 


341966 


32 


29 


617450 


4-62 


959081 


.96 


65836 9 


5 


58 


34i63i 


3 1 


3o 


617727 


4-62 


939023 


.96 


658704 


5 


58 


341296 


3o 


3i 


9-618004 


4-6i 


9-958965 


.96 


9*639039 
639373 


5 


58 


10-340961 


3 


3 2 


618281 


4-6i 


958908 


.96 


5 


5 7 • 


340627 


33 


6i8558 


4-6i 


95885o 


.96 


659708 


5 


5 7 


340292 


27 


34 


6i8834 


4-6o 


958792 
9^8734 


.96 


660042 


5 


57 


339968 


26 


35 


619110 


4-6o 


.96 


660376 


5 


5 7 


339624 


25 


36 


6i 9 386 


4- 60 


9 586 77 


.96 


660710 


5 


56 


339290 
338 9 57 


24 


n 


619662 


4- 5 9 


958619 


.96 


661043 


5 


56 


23 


6i 99 38 


4-5 9 


95856i 


.96 


661377 


5 


56 


338623 


22 


39 


620213 


4-5 9 


9585o3 


•97 


661710 


5 


55 


338290 


21 


4o 


620488 


4-58 


958445 


•97 


662043 


5 


55 


337967 


20 


4i 


9-620763 


4-58 


9-958387 


•97 


9-662376 


5 


55 


10-337624 


\l 


42 


62io38 


4-57 


958329 


•97 


662709 


5 


54 


337291 


43 


62i3i3 


4-57 


958271 


•97 


663o42 


5 


54 


336 9 58 


n 


44 


62i58 7 


4- 5 7 
4-56 


9582i3 


•97 


6633 7 5 


5 


54 


336625 


16 


45 


621861 


9 58 1 54 


•97 


663707 


5 


54 


336293 


i5 


46 


622i35 


4-56 


958096 


•97 


664039 


5 


5j 


335961 


14 


% 


622409 


4-56 


958o38 


•97 


664371 


5 


53 


335629 


i3 


622682 


4-55 


957979 


•97 


664703 


5 


53 


335297 


12 


49 


622956 


4-55 


957921 


•97 


665o35 


5 


53 


334965 


11 


5o 


623229 


4-55 


957863 


•97 


665366 


5 


52 


334634 


10 


5i 


9-6235o2 


4-54 


9-957804 


•97 


9*665697 


5 


52 


io-3343o3 


I 


52 


623774 


4-54 


957746 


.98 


666029 


5 


52 


333971 


53 


624047 


4-54 


957687 


.98 


66636o 


5 


5i 


333640 


7 


54 


624319 


4-53 


957628 


.98 


666691 


5 


5i 


333309 


6 


55 
56 


624D91 
624863 


4-53 

4-53 


957570 
95751 1 


:jS 


667021 
667352 


5- 
5 


5i 
5i 


332979 
332643 


5 

4 


tl 


625i35 


4-52 


957452 


.98 


667682 


5- 


5o 


3323i8 


3 


623406 


4-02 


957393 


.98 


668oi3 


5- 


5o 


331987 


2 


5, 


625677 
625948 


4-32 


957335 


.98 


668343 


5- 


5o 


33i657 
33i3a8 


1 


6o 


4-5i 


957276 


.98 


668672 


5-5o 







Cosine 


D- 


Sine 


65° 


Ootang. 


D. 


-J«ifc_ 


M, 



:nes and tangents. (2 t 5 degrees.) 



43 



o 


Sine 


1). 


Cosine 


D. 


Tung. 


Di 


Cotang. 




9-625948 


4-5i 


9-957276 


] -9« 


9.668673 


5-5o 


io-33i327 


60 


I 


6262ig 


4-5i 


957217 


.98 


669002 


5-49 


330998 


58 


2 


626490 


4-5i 


957168 


•9« 


669332 


5-49 


33o668 


3 


626760 


4-5o 


9 5 7°99 


.98 


669661 


5-49 


33o33 9 


5 7 


4 


627030 


4-5o 


957040 


.98 


669991 


5-48 


330009 


56 


5 


627300 


4-5o 


906981 


.98 


670320 


5-48 


329680 


55 


6 


627570 


4-49 


936921 


•99 


670649 


5-48 


329351 


54 


I 


627840 


4.49 


956862 


•99 


670977 
671306 


5-48 


329023 


53 


628109 


4.49 


9D6803 


•99 


5-47 


328694 


52 


9 


628378 


4-48 


956744 


•99 


671634 


5-47 


328366 


5i 


10 


628647 


4-48 


906684 


•99 


671963 


5-47 


328037 


5o 


ii 


9-628916 


4-47 


0-956625 


•99 0-672291 


5-47 


10.327709 


49 


12 


629185 


4-47 


9 56566 


•99 


672619 


5.46 


327381 


48 


i3 


629453 


4-47 


9565o6 


•'99 


672947 


5-46 


327053 


47 


14 


629721 


4-46 


956447 


•99 


673274 


5-46 


326726 


46 


i5 


629989 


4-46 


9 5638 7 


•99 


673602 


5.46 


3263 9 8 


45 


16 


63o2D7 


4-46 


956327 


•99 


673929 


5-45 


326071 


44 


»7 


63o524 


4.46 


956268 


•99 


674257 


5.45 


325743 


43 


18 


630792 


4-45 


956208 


1 -00 


674584 


5-45 


325416 


42 


»9 


63 1009 


4-45 


956148 


I -00 


674910 


5-44 


325090 


4i 


20 


63i326 


4-45 


956089 


I -00 


675237 


5-44 


324763 


40 


21 


9-63i5g3 


4-44 


9.956029 


I -00 


9 .675564 


5-44 


10. 324436 


39 


22 


63i85 9 
632125 


4-44 


955969 


I -00 


675890 


5-44 


324110 


38 


23 


4-44 


955909 


I -00 


676216 


5-43 


323784 


37 


24 


632392 


4-43 


955849 


I -00 


676543 


5-43 


323457 


36 


25 


632658 


4-43 


955789 


I -00 


676869 


5-43 


323i3i 


35 


26 


632923 


4-43 


955729 


I -00 


677-194 


5-43 


322806 


34 


27 


633189 


4-42 


955669 


I -00 


677520 


5-42 


322480 


33 


28 

2Q 


633454 
633719 


4-42 
4-42 


955609 

955548 


I -00 
I -00 


. 677846 
678171 


5-42 

5 • 42 


322i54 
321829 


32 

3i 


3o 


633 9 84 


4-4i 


900488 


1-00 


678496 


5-42 


32i5o4 


3o 


3i 


9-634249 


4-4i 


9-955428]! -oi 


9.678821 


5-4i 


ro .32ii79 


29 


32 


6345 1 4 


4.40 


955368,1-01 


679 1 46 


5-41 


320854 


28 


33 


634778 


4-4o 


955307 1 -oi 


679471 


5-4i 


32052O 


27 


34 


635o42 


4-4o 


955247 1 -oi 


679795 
6S0120 


5.41 


320203 


26 


35 


6353o6 


4-3 9 


905186 


I-OI 


5-4o 


319880 


25 


36 


635570 


4-3 9 


955i26 


I -01 


68o444 


5-4o 


319556 


24 


37 


635834 


4 i2 


955o65 


I -01 


680768 


5 • 40 


319232 


23 


38 


636o 97 


4-38 


955oo5 


I -01 


681092 


5-4o 


318908 


22 


3 9 


63636o 


4-38 


954944 


I ••01 


681416 


5-3 9 


3 1 8584 


21 


40 


636623 


4-38 


9 54883 


I -01 


681740 


5-3 9 


318260 


20 


41 


9-636886 


4-3 7 


9-954823 


I -01 


g. 682063 


5.39 


io . 317937 


3 


42 


637148 


4-3 7 


904762 


I -01 


682387 


5.39 
5-38 


317613 


43 


63741 1 


4-37 


954701 
904640 


I-OI 


6827IO 


317290 


17 


44 


63 7 6 7 3 


4-37 


I-OI 


683o33 


5-33 


( 316967 


16 


45 


637 9 35 


4-36 


954570 
904518 


1 -c: 


683356 


5-38 


3 1 6644 


i5 


46 


638197 
638458 


4-36 


I »02 


683679 


5-38 


3i632i 


14 


2 


4-36 


954457 


1-02 


684001 


5.37 


315999 


i3 


638720 


4-35 


954396 


1-02 


684324 


5.37 


315676 


12 


^ 9 


638 9 8i 


4-35 


954335 


1-02 


684646 


5.37 


3i5354 


11 


5o 


639242 


4-35 


954274 


I -02| 


684968 


5.37 


3i5o32 


10 


5i 


9-63 9 5o3 


4-34 


9-954213 


1-02 


9.680290 


5-36 


io-3i47io 


9 


52 


639764 


4-^4 


954152 


1-02 


6S56i2 


5-36 


3i4388 


8 


53 


640024 


4-34 


954090 


I -02! 


685 9 34 


5-36 


3i4o66 


7 


54 


640 2 84 


4-33 


954029 


1-02! 


6S6255 


5-36 


3i3745 


6 


55 


640544 


4-33 


953968 


1-02 


6865 77 


5-35 


3 1 3423 


5 


56 


640804 


4-33 


953906 


I -02 


6S6898 


5-35 


3i3io2 


4 


n 


641064 


4-32 


953845 


I -02 


6S7219 


5-35 


312781 


3 1 


641324 


4-32 


953783 


I -02 


687540 


5-35 


312460 


2 


5 9 


64i584 


4-32 


953722 


i-o3 


687861 


5-34 


3i 2i3q 
3ii8i8 


1 


60 


641842 


4-3i 


953660 


i-o3| 


688182 


5.34 





LJ 


Cosine 


D. 


Sine 


84°, 


Cotanc. 1 


1). 


Timg. 


M. 



13 



44 


(20 


DEGREES.) A 


rABLE OF LOGARITHMIC 




M. 




Sine 


D. 


Cosine 


D. 


Tung. 


I). 


Coteng. 


"1 


9-641842 


4 


3i 


9-953660 


1-03 


9-688182 


5-34 


io-3ii8i8l 60" 


i 


642101 


4 


3i 


933399 




o3 


688002 


5 


34 


3i 14981 5q 


2 


642360 


4 


3i 


953537 
953475 


I 


o3 


688823 


5 


34 


311177 


58 


3 


642618 


4 


3o 




o3 


689143 


5 


33 


310867 


57 


4 


642877 


4 


3o 


9534i3 




o3 


689463 


5 


33 


3 10537 


56 


5 


643i35 


4 


3o 


953352 




o3 


689783 


5 


33 


310217 


55 


6 


643393 


4 


3o 


903290 




o3 


690103 


5 


33 


309897 


54 


I 


648660 


4 


29 


q53228 




o3 


690423 


5 


33 


309677 


53 


643908 


4 


29 


<p3i66 




o3 


690742 


5 


32 


309208 


52 


9 


644i65 


4 


29 


963104 




o3 


691062 


5 


32 


3o8 9 38 


5i 


10 


644423 


4 


28 


903042 




o3 


69 1 38 i 


5 


32 


308619 


5o 


i i 


9 • 644680 


4 


28 


9-952980 




04 


9-691700 


5 


3i 


io-3o83oo 


49 


12 


644936 


4 


28 


952918 




04 


692019 


5 


3-1 


307981 


48 


i3 


645193 


4 


27 ■ 


952855 




04 


6 9 2338 


5 


3i 


307662 


47 


14 


640400 


4 


27 


902793 




04 


692656 


5 


3i 


3o 7 344 


46 


i5 


645706 


4 


27 


952731 




04 


692975 


5 


3i 


307025 


45 


16 


643962 


4 


26 


952669 1 1 


04 


693293 


5 


3o 


306707 


44 


\l 


646218 


4 


26 


962606 




04 


693612 


5 


3d 


3o6388 


43 


646474 


4 


26 


952544 




04 


693930 


5 


3o 


306070 


42 


19 


646729 


4 


25 


952481 




04 


694248 


5 


3o 


306752 


4i 


20 


646984 


4 


25 


902419 




04 


694666 


5 


29 


3o5434 


40 


21 


9-647240 


4 


25 


9.902356 




04 


9.694883 


5 


29 


io-3o5ii7 


?3 


22 


647494 


4 


24 


932294 




04 


696201 


5 


29 


304799 


38 


23 


647749 


4 


24 


95223i 




04 


695518 


5 


29 


304482 


37 


24 


648004 


4 


24 


952168 




o5 


6 9 5836 


5 


29 


304164 


36 


25 


648258 


4 


24 


902106 




o5 


696153 


5 


28 


3o3847 


35 


26 


648012 


4 


23 


952043 




o5 


696470 


5 


28 


3o353o 


34 


27 


648766 


4 


23 


961980 




o5 


696787 


5 


23 


3o32i3 


33 


28 


649020 


4 


23 


901917 




o5 


697103 


5 


28 


302897 


32 


29 


649274 


4 


22 


901854 




o5 


697420 


5 


27 


3o258o 


3i 


3o 


649627 


4 


22 


951791 


i 


o5 


697736 


5 


27 


302264 


3o 


3i 


9-649781 


4 


22 


9.951728 


' 


o5 


9.698053 


5 


27 


10.301947 


11 


32 


65oo34 


4 


22 


901660 


j 


o5 


698369 


5 


27 


3oi63i 


33 


630287 


4 


21 


901602 




o5 


6 9 8685 


5 


26 


3oi3i5 


27 


34 


65o539 


4 


21 


9§i539 




03 


699001 


5* 


26 


300999 


26 


35 


630792 


4 


21 


95i47 6 


1 


o5 


699316 


5 


26 


300684 


25 


36 


631044 


4 


20 


951412 


1 


o5 


699632 


5 


26 


3oo368 


24 


u 


651297 


4 


20 


95i 349 




06 


699947 


5 


26 


3ooo53 


23 


65 1 549 


4 


20 


951286 




06 


700263 


5 


23 


299737 


22 


39 


65 1 800 


4 


J 9 


951222 




06 


700678 


5 


23 


299422 


21 


40 


652o52 


4 


l 9 


95ii59 




06 


700893 


5 


25 


299107 


20 


41 


9-6523o4 


4 


J 9 


9.951096 




06 


9.701208 


5 


24 


10.298792 


:g 


42 


652555 


4 


18 


9610,32 




06 


70i523 


5 


24 


298477 


43 


65a8o6 


4 


18 


900968 




06 


701837 


5 


24 


2 9 8i63 


•7 


44 


653o57 


4 


18 


930906 




06 


702152 


5 


24 


297848 


16 


45 


6533o8 


4 


18 


900841 




06 


702466 


5 


24 


297334 


i5 


46 


65*3558 


4 


'7 


960778 




06 


702780 


5 


23 


297220 


14 


47 


6538o8 


4 


17 


950714 




06 


703095 


5 


23 


296905 


i3 


48 


634059 


4 


\l 


960660 




06 


703409 


5 


23 


296091 


12 


49 


654309 


4 


9 5o586 




06 


703723 


5 


23 


296277 


11 


30 


654558 


4 


16 


900622 




07 


704036 


5 


22 


296964 


10 


5i 


9-604808 


4 


16 


9.960458 




07 


9-7o435o 


5 


22 


10.293650 


I 


32 


666068 


4 


16 


900394 


, 


07 


704663 


5' 


22 


296337 


53 


605307 


4 


10 


93o33o 1 


07 


704977 


5 


22 


296023 


I 


54 


655556 


4 


13 


900266 1 


07 


706290 


5 


22 


294710 


55 


6558o5 


4 


1 5 


900202 1 


o.7 


700603 


5 


21 


294397 


5 


56 


656o54 


4 


14 


90O138 1 ! 


07 


700916 


5 


21 


294084 


4 


u 


656302 


4 


14 


900074 1 


07 


706228 


5 


21 


293772; 


65655i 


4 


14 


900010 




07 


706541 


5 


21 


293469! 2 


5 9 


656799 


4 


i3 


949945 


I 


"7 


706864 


5 


21 


293146I 1 


60 ! 657047 


4 


i3 


949881 


I-0 7 


707166 


5-20 


292834.I 




Cosiufc 


" 


D. 


Sine • 


6 


3° 


Cotang. 


D. 


Taiig. J M. J 



SINES AND TANGENTS. (27 DEGREES.) 



45 



f M. 


Sine 


D. 


Cosine | D. 


Tung. 


D. 


Cotang. 


I 


o 


9-65704- 


4-i3 


9-949881 I -O" 


9-707i6( 


). 5-20 


10-292834; 60 


[ i 


65 7 2 9 f 


4-i3 


949816:1 -0' 


7°747' 


I 5-20 


292522 1 5o 


1 I 


657542 


4-12 


949752|i-o- 


70779c 


) 5-20 


292210 58 


3 


65779c 
658o3i 


4-12 


94968811-0^ 


708 I OS 


5-20 


291898 


57 


4 


4-12 


949623 j 1 -ob 


70841^ 


5-19 


291586 


56 


5 


658284 


4-12 


949558' 1 -oE 


70872C 


5-19 


29127^ 


55 


6 


65853i 


4-u 


949494 ! i-oE 


70903- 


5-19 


29096c 


54 


I 


6D877S 


4-u 


949429 i- 08 


70934c, 


5-19 


290651 


53 


639025 


4- 11 


94936z 


ii-.oE 


70966c 


5-19 


290340 


52 


9 


' 659271 


4- 10 


94 9 3oc 


) 1 -oS 


709971 


5-i8 


290029 


5i 


IO 


659517 


4-io 


94923^ 


) 1 -08 710282 


5-i8 


289718 


5o 


ii 


9-659760 


4- 10 


9.94917c 


) i-oS 


9-71059,1 


5-i8 


10-289407 


% 


12 


660009 


4-09 


94910' 


» i-oS 


7 1 0904 


5-i8 


289096 


i3 


66o255 


4-09 


94904c 


1-08 


71121a 


5-i8 


288785 


47 


14 


66o5oi 


4*09 


94897: 


1-08 


7Il523 


5-17 


288475 


46 


i5 


660746 


4*09 


9489 1 c 


1-08 


7ii836 


5-17 


288164 


45 


16 


660991 


4-o8 


94884c 


1-08 


712146 


5.17 


2878^4 


44 


17 


66i236 


4-o8 


94878c 


1 -09 


712456 


5-17 


287544 


43 


18 


661481 


4-o8 


948 7 i£ 


1 -09 


712766 


5-i6 


287234 


42 


19 


661726 


4-07 


94865c 


1 -09 


713076 


5-i6 


286924 


41 


20 


661970 


4-07 


948584 


1 -09 


7i3386 


5-i6 


286614 


40 


21 


9-662214 


4-07 


9-9485ic 


1 -09 


9-713696 


5- 16 


io-2863o4 


% 


22 


662459 
662703 


4*07 


948454 


1 -09 


714005 


5-i6 


285995 


23 


4-o6 


948388 


1 -09 


7i43i4 


5-i5 


285686 


3 7 


24 


662946 


4-o6 


948323 


1-09 


714624 


5-i5 


2853 7 6 


36 


25 


663190 


4-o6 


948257 


1 -09 


714933 


5-i5 


285067 


35 


26 


663433 


4-o5 


948192 


1-09 


715242 


5-i5 


284758 


34 


27 


663677 


4-o5 


948126 


1-09 


7i555i 


5-i4 


284449 


33 


28 


663920 


4-o5 


948060 


1 -09 


7i586o 


5-i4 


284140 


32 


2 9 


664i63 


4-o5 


94799 5 


I -10 


716168 


5-i4 


283832 


3i 


3o 


664406 


4-o4 


947929 


I-IO 


716477 


5-i4 


283523 


3o 


3i 


9-664648 


4-o4 


9-947863 


I-IO 


9-716785 


5-i4 


io-2832i5 


3 


32 

33 

34 


664891 
665 1 33 


4-o4 


947797 
94773i 


I -10 


717093 


5-i3 


282907 


4-o3 


I -10 


7i74oi 


5-i3 


282599 


27 


665375 


4-o3 


947665 


I -10 


717709 


5-i3 


282291 


26 


35 


665617 


4-o3 


947600 


I-IO 


718017 


5-i3 


281983 


25 


36 


66585 9 


4-02 


947533 


I-IO 


7i8325 


5-i3 


281670 


24 


ll 


666100 


4-02 


947467 


I-IO 


7i8633 


5- 12 


28i36 7 


23 


666342 


4-02 


947401 


I-IO 


718940 


5-12 


281060 


22 


3 9 


666583 


4-02 


947335 


I-IO 


719248 


5-12 


280752 


21 


4o 


666824 


4-oi 


947269 


I-IO 


719555 


5-12 


280445 


20 


4i 


9-667065 


4-oi 


9-947203 


I-IO 


9-719862 


5-12 


io-28oi38 


!8 


42 


6673o5 


4-oi 


947i36 


I -II 


720169 


5. II 


279831 


43 


667546 


4-oi 


947070 


I'll 


720476 


5. II 


279524 


17 


44 


667786 


4-oq 


947004 


I'll 


720783 


5. II 


270217 
27891 1 


16 


45 


668027 


4-oo 


946937 
946871 


I'll 


721089 


5. II 


i5 


46 


668267 


4- 00 


I'll 


721396 


5. II 


278604 


14 


47 


6685o6 


^"99 


946804 


I'll 


721702 


5-io 


278298 


i3 


48 


668746 


f 99 


946738 


I'll 


722009 


5-io 


277991 


12 


i 9 


668986 


f 99 


946671 


I'll 


7223i5 


5-io 


277685 


11 


5e 


669225 


3 


946604 


I • 1 1 


722621 


5-io 


277379 


10 


5i 


9-669464 


9 946538 


I'll 


9-722927 


5-io 


10-277073 


I 


52 


669703 


3.98 


946471 


I'll 


723232 


5-09 


276768 


53 


669942 


3.98 


946404 


I'll 


723538 


5-09 


276462 


1 


54 


670181 


3 '91 


946337 


I'll 


723844 


5-og 


276156 


6 


55 


670419 


f'97 


946270 


I • 12 


724U9 


5-09 


2758D1 


5 


56 


670658 


3'97 


946203 


I-I2 


724454 


5-og 
5-o8 


275546 


4 


57 


670896 
671134 


3- 97 


946 1 36 


I-I2 


724759 


275241 3 


58 


3.96 


946069 


I-I2 


725o65 


5«o8 


274935 2 


£ 9 


671372 


3- 9 6 


946002 


I-I2 


725369 


5-o8 


27463 1 1 i 


60 


671609 


3- 9 6 


945935 


I • 12 


725674 


5-o8 


2743261 J 


1 Cosine 


D. 


Sine 


62° 


Cotang. 


D. 


Tang, j Ml 



46 


(28 DEGREES.) A TABLE OF LOGARITHMIC 


M. 


Sine 


1 D - 


Cosine D. 


Tang. 


1 D- 


Cotang. 


1 


o 


9*6716091 3-96 


0-945935 1 .12 


9.725674! 5-o8 


10-274326 


1 60 


i 


671847 


3. 9 3 


945868 1 • 1 2 


725979 


1 5-o8 


274021 


1 5o 
58 


a 


672084 


3- 9 5 


9458oo 1 • 1 2 


726284 


5.07 


273716 


3 


672321 


3. 9 5 


945733 1 • 12 


726588 


5.07 


273412 


5 7 

■56 


4 


672558 


3. 9 5 


945666 1 -i2 


726892 


! 5.07 


273108 


5 


672795 


3-94 


945598 I -12 


727197 


5.07 


272803 


55 


6 


673032 


3.94 


94553i 1. 12 


727501 


5-07 


272499 


54 


I 


673268 


3-94 


945464 i- 1 3 


727805 


5.06 


272195 


53 


6735o5 


3-94 


945396 1 - 1 3 


728109 


5-o6 


271891 


5a 


9 


6 7 3 7 4i 


3. 9 3 


94W28 i-ij 


728412 


5-o6 


271588 


5i 


IO 


673977 


3- 9 3 


945261 1 • 1 3 


728716 


5-o6 


271284 


5o 


ii 


9-674213 


3.93 


9-945193 1 • i3 


9.729020 


5- 06 


10-270980 


8 


12 


674448 


3-92 


945i25 1 • i3 


729323 


5-o5 


270677 


i3 


674684 


3-92 


945o58,i . i3 


729626 


5-o5 


270374 


47 


14 


674919 


3.92 


944990! 1. 1 3 


729929 


5-o5 


| 270071 


46 


i5 


675i55 


3.92 


944922 1. 13 
944854 Ii- 1 3 


73o233 


5-o5 


269767 


45 


16 


675390 


3-91 


73o535 


5-o5 


269465 


44 


\l 


675624 


3*91 


944786 1. 1 3 


73o838 


5-o4 


269162 


43 


675859 


3-91 


9447 1 8 : 1 • 1 3 


73u4i 


5-o4 


26885 9 


42 


19 


676094 


3-91 


94465o 1 - 13 


73i444 


5-o4 


268556 


41 


20 


676328 


3-90 


944582 i- 1 4 


731746 


5-o4 


268254 


40 


21 


9-676562 


3 90 


9-9445i4 i-i4 


9.732048 


5-04 


10-267952 


18 


22 


676796 


3.90 


944446 1 • 14 


73235i 


5-o3 


267649 


23 


677030 


3.90 


944377 1 -14 


7^2653 


5-o3 


267347 


37 


24 


677264 


3.89 


94430911-14 


732955 


5-o3 


267045 


36 


25 


677498 


3.89 


944241 I- 14 


733257 


5-o3 


266743 


35 


26 


677731 


3.89 


944172 1 -14 


733558 


5-o3 


266442 


34 


27 


677964 


3-88 


944104^.14 


73386o 


5-02 


266140 


33 


28 


678197 


3-88 


944o36Ji . 14 


734162 


5-02 


265838 


32 


29 


67843o 


3-88 


943967 1. 14 
943899 1. 14 


734463 


5-02 


265537 


3i 


3o 


678663 


3-88 


734764 


5-02 


265236 


3o 


3i 


9-678895 


3.87 


9>94383o 1 • 14 


9.735066 


5-02 


10-264934 


29 


32 


679128 


3.87 


943761 1. 14 


735367 


5-02 


264633 


28 


33 


679360 


3-87 


943693 i- 1 5 


735668 


5-oi 


264332 


27 


34 


679592 


3.87 


9436241 • i5 


735969 


5-oi 


26403 1 


26 


35 


679824 


3-86 


943555ji-i5 


736269 


5-oi 


263731 


25 


36 


68oo56 


3-86 


943486 1 -i5 


736570 


5-oi 


26343o 


24 


37 


680288 


3-86 


943417 1 -i5 


736871 


5-oi 


263129 


23 


38 


68o5i, 9 


3-85 


943348 ii- 1 5 


737171 


5-oo 


262829 


22 


3 9 


680750 


3-85 


943279 i- 15 


737471 


5- 00 


262529 


21 


4o 


680982 


3-85 


94321011 -i5 


737771 


5-oo 


262229 


20 


4i 


9.681213 


3-85 


9-943141 ! 1 • i5 


9-738071 


5-oo 


10-261929 


19 


42 


68U43 


3.84 


943072J1 ..i5 


738371 


5-oo 


261629 


18 


43 


681674 


3-84 


943oo3|i-i5 


738671 


4-99 


261329 


17 


44 


68i 9 o5 


3.84 


942934 i- i5 


7 38 97 i 


4.99 


261029 


16 


45 


682i35 


3-84 


942864! i- 15 


739271 


4.99 


' 260729 


i5 


46 


682365 


3-83 


942795 i- 16 


739570 


4-99 


26o43o 


14 


% 


6825 9 5 


3-83 


94272611-16 


739870 


4-99 


26oi3o 


i3 


682825 


3-83 


942656 i- 16 


740169 
740468 


4-99 


25 9 83i 


12 


49 


683o55 


3-83 


942587 i- 16 


4.98 


259532 


1 1 


5o 


683284 


3.82 


942517 1-16 


740767 


4.98 


s5o233 
10-258934 


10 


5i 


9-6835i4 


3.82 j 


9-942448 i- 16 


9.741066 


4.98 


9 


52 


683 7 43 


3.82 


94237811-16 


74i365 


4-98 


258635 


8 


53 


•683972 


3-82 


9423o8 i- 16 


741664 


4- 9 8 


258336 


7 


54 


684201 


3-8i 


942239 1 • 16 


741962 


4-97 


258o38j 6 


55 
56 


68443o 
684658 


3-8i 
3-8i 


942169 i- 16 
942099 i- 16 


742261 
742559 


4-97 

4-97 


257739 5 
257441! 4 


u 


684887 


3.8o 


9420291-16 


742858 


4-97 


257142! 3 


685u5| 


3-8o 


941959 1 -16 
941889 1-17 


743 1 56 


4-97 


256844 2 


59 


685343 


3-8o 


743454 


4-97 
4.96 

D. 


256546) 1 


60 


685571 


3.8o 


941819:1-17 


743752 
Cotang. 


2562481 




Cosine 


D. 


Sine 61° 


Tang. 1 


M.I 



BINES AND TANGENTS. (29 DEGREES.) 



47 



M. 


Sine 


1). 


Cosine | D. 


Tang. 


D. 


Cotang. | 





9-685571 


3-8o 


9-941819 1 -17 


9-743752 


4.96 


lo-256248| 60 


i 


685799 


3.79 


9417491-17 


744o5o 


4.96 


255960 5o 
255652! 58 


2 


686027 


3-79 


941679 1. 1 7 


744348 


4-96 


3 


686254 


3-79 


941609 1 • 17 


744645 


4.96 


255355! 57 


4 


686482 


$% 


94i53g 1. 17 


744943 


4-96 


255o57 56 


5 


686709 


941469 1 -17 


745240 


4.96 


264760; 55 


6 


686 9 36 


3-78 


941398 1 '17 


745538 


4-9 5 


2 5446 2 1 54 


7 


687 r 63 


3- 7 8 


94i328 1-17 


745835 


4-95 


264166 


1 53 


8 


687389 


3-78 


941258 1 • 17 


746i32 


4-95 


253868 


| 52 


9 


687616 


3-77 


941187 1 • 17 


746429 


4-95 


253571 


5i 


10 


687843 


3-77 


94iH7 ; i-i7 


746726 


4-9^ 


253274 


5o 


ii 


9-688069 


l"" 


9.941046^-18 


9-747023 


4-94 


10-252977 


49 


12 


688293 


3-77 


94oq75|i-i8 


747319 


4-94 


262681 


48 


i3 


688521 


3-76 


94oqo5|i-i8 


747616 


4-94 


252384 


47 


14 


688747 


3^6 


940834! i- 18 


7479 J 3 


4-94 


252087 


46 


i5 


688972 


3- 7 6 


q4o763,'i-i8 


748209 
7485o5 


4-94 


261791 


45 


16 


689198 


3.76 


94069311 -18 


4-93 


261495 


44 


17 


689423 


3-75 


940622J1 -18 


748801 


4-93 


251199 


43 


18 


689648 


3.75 


94o55i i- 18 


749097 


4-9 3 


260903 


42 


19 


689873 


3- 7 5 


940480 i- 1 8 


7493o3 
749689 


4-93 


260607 


4i 


20 


690098 


3. 7 5 


940409 i- 18 


4-93 


25o3n 


4o 


21 


9 -6903 23 


3-74 


9-94o338 1 -18 


9-749980 


4-93 


10-260016 


39 


22 


690548 


3*74 


940267 i- 18 


760281 


4-92 


249719 


38 


23 


690772 


3-74 


940196 i- 18 


760676 


4-92 


249424 


37 


24 


690996 


3-74 


940125 1-19 


750872 


4-92 


249128 


36 


25 


691220 


3- 7 3 


940054 1-19 


751167 


4-92 


248833 


35 


26 


691444 


3- 7 3 


939982 1-19 


751462 


4-92 


248538 


34 


27 


691668 


3.73 


939911 


1-19 


751757 


4-92 


248243 


33 


28 


691892 


3- 7 3 


9 3 9 8 4 o 


1-19 


762062 


4-9 1 


247948 


32 


29 


6921 1 5 


3. 7 2 


939768 


1-19 


762347 


4-91 


247653 


3i 


3o 


692339 


3.72 


939697 


1-19 


/52642 


4-91 


247358 


3o 


3i 


9-692562 


3-72 


9-939625 


1. 19 


9-752937 


4-9 1 


10-247063 


29 


32 


692785 


3.71 


939554 


1-19 


75323i 


4-9 1 


246769 


28 


33 


693008 


3.71 


93948211-19 


753526 


4-91 


246474 


27 


34 


693231 


3-71 


919410 1-19 


753820 


4.90 


246180 


26 


35 


693453 


3- 7 i 


939339 


1-19 


7641 i5 


4-90 


245885 


25 


36 


693676 


3-70 


939267 


1-20 


754409 


4-9° 


245591 


24 


37 


693898 


3-70 


939195 I -20 


754703 


4.90 


245297 


23 


38 


694120 


3-70 


939123 I- 20 


724997 


4-9° 


245oo3 


22 


3 9 


694342 


3-70 


93oo52 i- 20 
93098011- 20 


736291 

755585 


4.90 


244709 


21 


4o 


694564 


3-6 9 


4.89 


244416 


20 


4i 


9-694786 


3-6 9 


9-938908 1-20 


9-755878 


4-8 9 


10-244122 


\l 


42 


695007 


3.69 


9 38836 1-20 


706172 


4-89 


243828 


43 


696229 


3.69 
3-63 


9 38 7 63 1-20 


756465 


4-89 


243535 


n 


'44 


695430 


938691 


1 -20 


766759 


4-8 9 


243241 


16 


45 


695671 


3-68 


938619 


1-20 


757052 


4-89 
4-88 


242948 


i5 


46 


695892 


3-68 


938547 


1-20 


767346 


242655 


14 


47 


6961 1 3 


3-68 


938475 


1-20 


767638 


4-88 


242362 


i3 


48 


6 9 6334 


3-67 


938402 


I-2I 


75 79 3 1 


4-88 


242069 


12 


49 


6 9 6554 


3.67 


9 3833o 


I-2I 


768224 


4-88 


241776 


11 


5o 


696775 


3-67 


9 3S 2 58 


I-2I 


7585i7 


4-88 


• 241483 


10 


5i 


9-696995, 


3-67 


9- 9 38i85 


I -21 


9-7588io 


4-88 


10-241190 





52 


697215 


3-66 


9 3Sn3 


I- 21 


769102 


4-87 


240898 


8 


53 


697435! 


3-66 


938040 


I-2I 


769396 

769687 


4-87 


24o6o5 


7 


54 


6 97 654! 


3-66 


937967 


I - 21 


4-87 


24o3i3 


6 


55 


697874! 


3-66 


937895 


I -21 


759979 


4-8 7 


240021 


5 


56 


698094 


3-65 


937822 


I-2I 


760272 


4-87 


239728 


4 


5 7 


6 9 83 1 3 


3-65 


9 3 7749 


I -21 


760664 


4-87 


239436 


3 


58 


698532 


3-65 


937676 


I -21 


76o856 


4-86 


23qi44 


2 


5q 


698701 i 


3-65 


937604 


I -21 


761 148 


4-86 


238852 


1 


6o | 


69897c 
Cosine 1 


3.64 


937531 


1*21 


761439 


4-86 


23856i 
~TangT~ 





i 


D. i 


Sine 


60° 


Cotang. 


D. 



48 


(30 


DEGREES.) A 


rABLE OF LOGARITHMIC 




M. 


Sme 


D. 


Cosine | I). 


Tang. 


1). 


Cotang. 


*" 





9-698970 


3 


64 


9-937531 1 • 21 


9-761439 


4-86 


io-23856~i 


60 


i 


699189 


3 


• 64 


937458 1 


2 2 


761731 


4 


86 


238269 


it 


2 


699407 


3 


•64 


937385 1 


■22 


762023 


4 


86 


237977 


3 


699626 


3 


• 64 


9 3 7 3i2 1 


22 


762314 


4 


86 


237686 


5 7 


4 


699844 


3 


63 


937238 1 


22 


762606 


4 


85 


237394 


56 


5 


700062 


3 


63 


937165 1 


■22 


762897 
763188 


4 


85 


237103 
236812 


55. 


6 


700280 


3 


63 


937092 1 


22 


4 


85 


54 


I 


700498 


3 


63 


937019 




22 


7634-9 


4 


85 


236521 


53 


700716 


3 


63 


936046 
936872 




•22 


763770 


4 


85 


23623o 


52 


9 


700933 


3 


62 




22 


764061 


4 


85 


235939 
235648 


5i 


10 


701 i5i 


3 


62 


936799 




•22 


764352 


4 


84 


5o 


n 


9-701368 


3 


62 


9- 9 36725;i 


■22 


9-764643 


4 


84 


io. 2 3535 7 


49 


12 


7oi585 


3 


62 


936652,1 


23 


764933 


4 


84 


235067 


48 


i3 


701802 


3 


61 


936578: 1 


23 


765224 


4 


84 


234776 


47 


14 
i5 


702019 
702236 


3 
3 


61 
61 


9 365o5 1 

93643i 1 
936357! 1 


23 
•23 


7655i4 
7 658o5 


4 
4 


84 
84 


234486 
234195 


46 
45 


16 


702452 


3 


61 


•23 


766095 


4 


84 


233 9 o5 


44 


17 


702669 


3 


60 


936284 1 


•23 


766385 


4 


83 


2336i5 


43 


18 


702885 


3 


60 


9362io!i 


23 


766675 


4 


83 


233325 


42 


19 


7o3ioi 


3 


60 


936i36 1 


23 


766965 


4 


83 


233o35 


41 


20 


703317 


3 


60 


936062,1 


23 


767255 


4 


83 


232745 


40 


21 


9>7o3533 


3 


5 9 


9-935988 1 

935qi4i 


23 


9-767545 


4 


83 


io-232455 


3 9 


22 


703749 


3 


5 9 


23 


767834 


4 


S3 


232166 


38 


23 


703964 


3 


5 9 


930840 1 


23 


768124 


4 


82 


231876 


37 


24 


704179 


3 


i 9 


935766 




24 


768413 


4 


82 


23i587 


36 


25 


704395 


3 


ll 


935692 




24 


768703 


4 


82 


231297 


35 


26 


704610 


3 


9356i8 




24 


768992 


4 


82 


23 1 008 


34 


27 


704825 


3 


58 


935543 




24 


769281 


4 


82 


230719 


33 


28 


7o5o4o 


3 


58 


935460! 1 


/ 


769570 


4 


82 


23o43o 


32 


^ 9 


7o5254 


3 


58 


935395 




24 


769860 


4 


81 


23oi4o 


3i 


3o 


705469 


3 


57 


935320 




24 


770148 


4 


81 


22QS52 


3o 


3i 


9 -705683 


3 


57 


9-935246 




24 


9.770437 


4 


81 


io. 2 2 9 563 


3 


32 


705898 


3 


5 7 


93517 1 




24 


770726 


4 


81 


229274 


33 


706112 


3 


5 7 


935097 




24 


771015 


4 


81 


228985 


27 


34 


706326 


3 


56 


935022 




24 


77i3o3 


4 


81 


228697 


26 


35 


706539 


3 


56 


934948 




24 


77i592 
. 771880 


4 


81 


228408 


25 


36 


706753 


3 


56 


934873 




24 


4 


80 


228120 


24 


37 


706967 


3 


56 


934798 




25 


772168 


4 


80 


227832 


23 


38 


707180 


3 


55 


934723 




25 


772457 


4 


80 


227543 


22 


3 9 


707393 


3 


55 


934649 




2 5 


772745 


4 


80 


227255 


21 


4o 


707666 


3 


55 


934574 




25 


773o33 


4 


80 


226967 


20 


4i 


9.707819 


3 


55 


9-934499 




25 


9.773321 


4 


80 


10-226679 


!! 


42 


708032 


3 


54 


934424 




25 


773608 


4 


79 


226392 


43 


708245 


3 


54 


934349 




25 


77 38 9 6 


4 


79 


226104 


17 


44 


708458 


3 


54 


934274 


1 


25 


774i84 


4 


79 


2258i6 


16- 


45 


70S670 


3 


54 


934199 1 


25 


774471 


4 


79 


225529 


i5 


46 


708882 


3 


53 


934123 1 


25 


774759 


4 


79 


225241 


14 


3 


709094 


3 


53 


934048 1 


25 


775046 


4 


79 


224954 


i3 


709306 


3 


53 


933973 1 


25 


775333 


4 


79 


224667 


12 


49 


709518 


3 


53 


9338981 


26 


775621 


4 


78 


224379 


11 


5o 


709730 


3 


53 


933822 1 


26 


77 5 9 o8 


4 


78 


224092 


10 


5i 


9-70994I 


3 


52 


9'933747 




26 


9.776195 


4 


78 


io. 2 23So5 


3 


52 


7 1 1 53 


3 


02 


933671 




26 


776482 


4 


78 


2235i8 


53 


710364 


3 


52 


9 335 9 6 1 


26 


776769 


4 


78 


223231 


7 


54 


710575 


3 


52 


9 33520i 


26 


777055 


4 


78 


222945 


6 


55 


710786 


3 


5i 


933445 1 


26 


777342 


4 


/8 


222658 


5 


56 


710997 


3 


5i 


9333691 
933293I 1 


26 


777628 


4 


77 


222372 


4 


ll 


71 1208 


3 


5i 


26 


777915 


4 


77 


2220S5 


3 


711419 


3 


5i 


933217 i 


26 


778201 


4 


77 


221799 


2 


5 9 


711629 


3 


5o 


933141 1 


26 


778487 


4 


77 


22l5l2 


1 


6o 


71 1839 


3. 


5o 


933o66|i 


26 


778774 


4-77 


221226 

~Tan«. — 







Coeine 


D. 


Sine 1 59° 


Cotang. 


I 


). 



SINES AND TANGENTS. (31 DEGREES.) 



49 



k. 

o 


i Sine 


D. 


Cosine | D. 


Tang. 


D. 


Cotang. 




9-711839 


3-5o 


9-933066 1 -26 


9-77877^ 


4-77 


10-221226 


60 


I 


712060 


3 


.00 


932990! 1 -27 


779060 


4-77 


220940 


£ 


2 


712263 


3 


-5o 


93291411-2- 


779346 


4-76 


220654 


3 


712469 


3 


•49 


932838 


,1.27 


779632 


4-76 


22o368 


5 7 


4 


712679 


3 


•49 


932762 


L27 


779918 


4-76 


220082 


56 


5 


712889 


3 


•49 


9 3 2 685 


1.27 


780203 


4-76 


219797 


55 


6 


713098 


3 


•49 


932609 


11-27 


780489 
780775 


4-76 


219511 


54 1 


7 


7i33o8 


3 


• 49 


932533 


1.27 


4-76 


219220 


53 


8 


7i35i 7 


3 


.48 


932457 


1.27 


781060 


4-76 


' 2 1 8940 


52 


9 


713726 


3 


.48 


93238o 


1.27 


7 8i346 


4-75 


2i8654 


5i 


10 


713935 


3 


•48 


g323o4 


1.27 


78i63i 


4-75 


2i836g 


5o 


n 


9-714144 


3 


• 48 


9-932228 


1.27 


9-781916 


4-75 


10-218084 


49 


12 


714352 


3 


47 


932i5i 


1-27 


782201 


4-75 


217799 


48 


i3 


i456i 


3 


47 


932075 


1-28 


782486 


4-75 


217514 


47 


U 


7U769 
714978 


3 


47 


931998 


1-28 


782771 


4-75 


217229 


46 


i5 


3 


47 


931921 


1-28 


783o56 


4-75 


216944 


45 


16 


7i5i86 


3 


47 


9 3i845 


1-28 


783341 


4-75 


216659 


44 


17 


7 1 53 9 4 


3 


46 


9 3i 7 68 


1.28 


783626 


4-74 


216374 


43 


iS 


7i56o2 


3 


46 


931691 


1-28 


783910 


4-74 


216090 


42 


19 


715809 


3 


46 


931614 


1-28 


784195 


4-74 


2i58o5 


41 


20 


716017 


3 


46 


93i 537 


1.28 


784479 


4-74 


2 1 552 1 


4o 


21 


9-716224 


3 


45 


g-93i46o'i -28 


9-784764 


4-74 


io-2i5236 


ll 


22 


716432 


3 


45 


9 3i3S3,i-28 


7 85o48 


4-74 


214952 


23 


716639 


3 


45 


93i3o6 1 -28 


785332 


4-73 


214668 


37 


24 


716846 


3 


45 


9 3 1 229:1 -29 


7856i6 


4-73 


214384 


36 


25 


717053 


3 


45 


93 1 1 J2 1 -29 
93io75|i -29 
930998 1.29 


785900 


4-73 


214100 


35 


26 


717259 


3 


44 


7S6184 


4-73 


2i38i6 


34 


27 


717466 


3 


44 


786468 


4-73 


2i3532 


33 


28 


717673 


3 


44 


930021 1-29 


706752 


4-73 


213248 


32 


29 


717879 


3 


44 


930843^-29 


707036 


4- 7 3 


2 1 2964 


3i 


3o 


718085 


3 


43 


930766 1-29 


7G7319 


4-72 


2 1 2681 


3o 


3i 


9-718291 


3 


43 


9.930688 


1.29 


9-787603 


4-72 


10-212397 


3 


32 


718497 


3 


43 


93o6i 1 


I -29 


787S86 


4-72 


21 2 1 14, 


33 


718703 


3 


43 


93o533 


1.29 


788170 


4-72 


2ii83o 


27 


34 


718909 


3 


43 


93o456;i -29 


788453 


4-72 


211547 


26 


35 


7I9H4 


3 


42 


93,378 


1-29 


788736 


4-72 


21 1264 


25 


36 


719320 


3 


42 


93o3oo 


i-3o 


789019 


4-72 


2 1 098 1 


24 


37 


7i 9 525 


3 


42 


930223 


i-3o 


789302 


4-71 


210698 


23 


38 


719730 


3 


42 


93oi45 


i-3o 


78 9 585 


4- 7 t 


2io4i5 


22 


39 


7'9935 


3 


4i 


930067 


i-3o 


789868 


4- 7 i 


2!Ol32 


21 


4o 


720140 


3 


4i 


929989 


i-3o 


7901 5 1 


4- 7 i 


209S49 


20 


4i 


9-72o345 


3 


4i 


9-92Q9U 


i-3o 


•9-790433 


4- 7 i 


10-209067 


!2 


42 


720549 


3- 


4i 


929833 


i-3o 


•790716 


4-7i 


209284 


43 


720754 


3- 


40 


929755 


i-3o 


790999 


4-71 


209001 


17 


44 


720958 


3- 


40 


929677 


i-3o 


791281 


4-71 


208719 


16 


45 


721162 


3- 


40 


92 9 5 99 


i-3o 


79 1 563 


4-70 


208437 


1 5 


46 


72i366 


3- 


40 


929021 


i-3o 


791846 


4-73 


208 1 54 


14 


47 


721570 


3- 


4o 


929442 


i-3o 


792128 


4-70 


207872 


i3 


48 


72H74 


3- 


39 


929364 


1 -3i 


792410 


4-70 


207590 


12 


P 


721978, 


3- 


3 9 


929286 


i-3i 


792692 


4-70 


207308 1 1 


5o 


722181 


3- 


39 


929207 


1. 3 1 


792974 


4-70 


207026, 10 


5i 


9-722385: 


3- 


39 


9.929129 


1 -3i 


9-793256 


4-70 


10-206744 9 


52 


722088 


3- 


u 


929050 


i-3i 


793538 


4-6 9 


206462 8 


53 


722 791 j 


3- 


928972 


i-3i 


793819 


4-69 


2061S1 | 7 


54 


722994, 


3- 


38 


928893 j 1. 3 1 


794ioi 


4-6 9 


205899^ 6 


55 


723197 


3- 


38 


928815,1 - 3 1 j 


794383 


4-6 9 


205617 5 


56 


723400 


3- 


38 


928736 i-3i! 
928657I1.31 


794 66 4 


4-6 9 


205336 4 


57 


7236o3 : 


3- 


37 


79-4945 


4-69 


2o5o55J 3 


58 


7238o5l 


3- 


37 


928578:1.311 


795227 


4-6o 
4-68 


204773J 2 


J 9 


724007 


3- 


37 


928499 1 -3i | 


7955o8 


204492 I 


60 


724210 


3.37 


928420 1 -3i| 


7 9 5 7 8 9 


4-68 


2042 m 


Cosine 


1). 


Sine |58 o1 


"Cotancr. 


D. 


Tang. 1 


_M^ 



50 


(32 DEGREES.) A 


TABLE OF LOGARITHMIC 




M. 


Sine 


D. 


Cosine | D. 


Tang. 


D. 


Cotang. 







9-724210 


3-37 


9-9284201-32 


9.795789 


4-68 


10-204211 


60 


i 


724412 


3 


•3 7 


928342! I -32 


796070 


4-68 


203930 


u 


2 


724614 


3 


•36 


928263:1-32 


796351 


4-68 


203649 


3 


724816 


3 


-36 


9 28i83|i-32 


796632 


4-68 


203368 


57 


4 


725017 


3 


•36 


928104' I -32 


796913 


4-68 


203087 


56 


5 


725219 


3 


36 


928025 I -32 


797194 


4-68 


202806 


55 


6 


725420 


3 


35 


927946 I -32 
927867J1 -32 


797475 


4-68 


202525 


54 


I 


725622 


3 


35 


797755 


4-68 


202245 


53 


725823 


3 


35 


92778-7I I -32 


798036 


4-67. 


201964 


52 


9 


726024 


3 


35 


927708 


1-32 


7 9 83 1 6 


4-67 


20I684 


5i 


10 


726225 


3 


35 


927629 


1-32 


798596 


4-67 


201404 


5o 


ii 


9-726426 


3 


34 


0-927549 


1-32 


9.798877 


4-67 


IO-20II23 


49 


12 


726626 


3 


34 


927470 


1-33 


799157 


4-67 


200843 


48 


i3 


726S27 


3 


34 


927390 


i-33 


799437 


4-67 


2oo563 


47 


U 


727027 


3 


34 


927310 


i-33 


799717 


4-67- 


200283 


46 


i5 


727228 


3 


34 


927231 


i-33 


799997 


4-66 


200003 


45 


16 


727428 


3 


33 


92701 


1-33 


800277 


4-66 


199723 


44 


17 


727628 


3 


33 


927071 


i-33 


8oo557 


4-66 


199443 


43 


18 


727828 


3 


33 


926991 


i-33 


Goo836 


4-66 


199164 


42 


! 9 


728027 


3 


33 


92691 1 


1-3.3 


801 1 16 


4-66 


198884 


4i 


20 


728227 


3 


33 


926831 


i-33 


C01396 


4-66 


198604 


40 


21 


9-728427 


3 


32 


9-926751 


i-33 


9-801675 


4-66 


io- 198325 


% 


22 


72S626 


3 


32 


92667 1 


1-33 


8oi 9 55 


4-66 


198045 


23 


723825 


3 


32 


926591 


i-33 


802234 


4-65 


197766 


37 


24 


729024 


3 


32 


9265ii 


i-34 


8 02 5i3 


4-65 


197487 


36 


25 


729223 


3 


3i 


926431 


i-34 


802792 


4-65 


197208 


35 


26 


729422 


3 


3i 


92635i 


i-34 


803072 


4-65 


196928 


34 


27 


729621 


3 


3i 


926270 


i-34 


8o335i 


4-65 


196649 


33 


28 


729820 


3 


3i 


926190 


1-34 


8 363o 


4-65 


196370 


32 


29 


730018 


3 


3o 


9261 10 


i-34 


803908 


4-65 


196092 


3i 


3o 


730216 


3 


3o 


926029 


i-34 


804187 


4-65 


195813 


3o 


3i 


9- 7304 1 5 


3 


3o 


9-925949 


i-34 


9-804466 


4-64 


io- 195534 


3 


32 


73o6i3 


3 


3o 


925868 


i-34 


8o4745 


4-64 


195255 


33 


73o8i 1 


3 


3o 


925788 


i-34 


8 5o23 


4-64 


194977 


27 


34 


731009 


3 


29 


925707 


i-34 


8o53o2 


4-64 


194698 


26 


35 


731206 


3 


29 


920626 


i-34 


8o558o 


4-64 


194420 


25 


36 


73 1 404 


3 


29 


925545 


i-35 


8o585 9 


4-64 


194141 


24 


37 


731602 


3 


29 


925465 


1 -35 


806137 


4-64 


193863 


23 


38 


731799 


3 


29 


925384 


1 -35 


8064 1 5 


4-63 


i 9 3 585 


22 


3 9 


731996 


3 


28 


9253o3 


1 -35 


8o66 9 3 


4-63 


193307 


21 


4o 


73-2I93 


3 


28 


925222 


1 -35 


806971 


4-63 


193029 


20 


41 


9-732390 


3- 


23 


9-925i4i 


1 -35 


9-807249 


4-63 


io- 192751 


'9 


42 


732587 


3 


28 


925060 


1 -35 


807527 


4-63 


192473 


18 


43 


732784 


3- 


23 


9249*79 


i-35 


807805 


4-63 


192195 


'7 


44 


732980 


3- 


27 


924897 


i-35 


8o8o83 


4-63 


>9i9i7 


16 


45 


733177 


3 


27 


924816 


i-35 


8o836i 


4-63 


191639 


i5 


46 


733373 


3- 


27 


924735 


i-36 


8oS633 


4-62 


191362 


14 


47 


733569 


3- 


27 


924654 


i-36 


808916 


4-62 


191084 


i3 


48 


733765 


3- 


27 


924572 


i-36 


809193 


4-62 


1 90807 | 12 


49 


733961 


3- 


26 


924491 


i-36 


809471 


4-62 


190529J ii 


5o 


734i5 7 


3- 


26 


9 24409 | 
9-924328 


i-36 


809748 


4-62 


190252 10 


5i 


9-734353 


3- 


26 


i-36 


9-810025 


4-62 


io- 189975 


i 


52 


734549 


3- 


26 


924246, 


i-36 


8io3o2 


4-62 


189698 


53 


734744 


3- 


25 


924164; 


i-36 


8io58o 


4-62 


189420 


7 


54 


. 734939 


3- 


25 


924083 


i.36 

1 -36, 


810857 


4-62 


189143I 6 
1 888661 5 


55 


735i35 


3- 


25 


924001 


8in34 


4- 61 


56 


73533o 


3- 


25 


923919 1 


i-36! 


811410 


4-6i 


1 885 Q o 4 


57 


735525 


3- 


25 


923837! 1 -36 


81 1687 


4-61 


i883i3! 3 


58 


735719 


3- 


24 


923755,1 -371 


81 1964 


4-61 


i88o36, 2 


5 9 


733914 


3- 


24 


923673 


i-3 7 


81 2241 


4-6i 


187759 1 | 


60 


736109 


3- 


24 


923591 


i.3 7 | 
57° 


812517 


4-6i 


1 8748I. 


1 


Cosine 


T). 


Sine 


Cotang. 1 


D. 


"Tim?. ' 





8INEE 


! AND TANGENTS. 


(33 DEGREES. 1 


> 


51 


Ml 


Sine 


D. 


Cosine | D. 


Tang. 


D. 


Cotang. 







9-736109 


3 


•24 


9-923591 1 -37 


9-812517 


4-6i 


10-187482 


~6o~ 


i 


7363o3 


3 


•24 


923509 1 


•3 7 


812794 


4-6i 


187206 


5o 


«j 


736498 


3 


■24 


923427 1 


•37 


8 1 3070 


4-61 


i86 9 3o 


58 


3 


736692 


3 


•23 


923345 1 


•37 


813347 


4- 60 


186653 


57 


4 


736886 


3 


•23 


923263 1 


•3 7 


0i3623 


4-6o 


1 S63 77 


56 


5 


737080 


3 


23 


923i8i 1 


• 3 7 


8 1 38 99 
8i4n5 


4-6o 


186101 


55 


6 


737274 


3 


•23 


923098 1 


•3 7 


4-6o 


185825 


54 


I 


737467 


3 


23 


923016 1 


• 3 7 


8i4452 


4- 60 


IS5548 


53 


737661 


3 


•22 


922933 1 


37 


814728 


4.60 


1S5272 


5? 


9 


737855 


3 


•22 


9228511 


37 


8i5oo4 


4-6o 


134906J 5 1 


IO 


738048 


3 


22 


922768 1 


-38 


816279 
9-8i5555 


4-6o 


1347211 5o 


1 II 


9-738241 


3 


22 


9-922686 1 


-38 


4-5 9 


10-184445 


8 


12 


738434 


3 


22 


922603 1 


• 38 


8i583i 


4-5 9 


1S4169 


i3 


738627 


8 


21 


922520 1 


• 38 


816107 


4-5 9 


i338 9 3 


47 


14 


738820 


3 


21 


922438 1 


38 


8i6382 


4 ■ 5c; 


iS36i8 46 


i5 


73gor3 


3 


21 


922355 1 


38 


8i6658 


4-59 


i83342| 45 


16 


739206 


3 


21 • 


922272:1 


38 


816933 


4-5 9 


183067 


44 


H 


73o3 9 8 


3 


21 


922189'! 


38 


817209 


4- 5 9 


182791 


43 


18 


739590 
73 97 83 


3 


2Q 


922106' I 


38 


817484 


4-5 9 


i325i6 


42 


l 9 


3 


20 


922023jl 


38 


817759 
8i8o35 


4-5 9 


182241 


41 


20 


739975 


3 


20 


921040:1 
9.921857 1 


38 


4-53 


i8i 9 65 


4o 


21 


9-740167 


3 


20 


39 


9-8i83io 


4-53 


16-181690 


3 9 


22 


74o359 


3 


20 


921774 1 


3 9 


8i8585 


4-53 


i3i4i5 


38 


23 


74o55o 


3 


'9 


921691I1 


3 9 


818860 


4-58 


181 140 


37 


•24 


740742 


3 


'9 


921607 1 


3g 


819135 


4-58 


i8o865 


36 


25 


74og34 


3 


19 


92i524 1 


39 


819410 


4-58 


i°o590 


35 


26 


74H25 


3 


'9 


92i44i|i 


39 


819684 


4-53 


i8o3i6 


34 


27 


74i3i6 


3 


•9 


921357 1 


39 


819959 


4-53 


180041 


33 


28 


741 5o8 


3 


18 


921274 1 


3o 


820234 


4-58 


179766! 3: 


29 


741699 


3 


18 


921190 1 


3 9 


82o5o8 


407 


179492! 3i 


3o 


741889 


3 


18 


921107 1 


3o 


820783 


4-57 


179217I 3o 
10-1789431 29 ! 


3i 


9-74208*0 


3 


18 


9-921023,1 


39 


9-821057 


4-57 


32 


742271 


3 


18 


920939,1 
92o356 ! i 


40 


82i332 


4-57 


178668 28 [ 


33 


742462 


3 


17 


4o 


821606 


4.57 


178394I 27 | 


34 


742652 


3 


'7 


920772^ 


40 


821880 


4-5 7 


1 73l 20! 26 


35 


742842 


3 


17 


920688 1 


40 


822154 


4-57 


177846 25 


36 


743o33 


3 


17 


920604 1 


40 


822429 
82270J 


4-5 7 


I77571I 1\ 


37 


743223 


3 


'7 


920520|I 


40 


407 


1772971 23 


38 


7434i3 


3 


16 


920436 I 


40 


822977 


4-56 


177023! 22 


3 9 


7436o2 


3 


16 


920352:1 


40 


828250 


4-56 


I76750 21 


4o 


743792 


3 


16 


920268 I 


40 


823524 


4-56 


1-76476 2() 


4i 


9-713982 


3 


16 


9-92018411 


40 


9-823798 


• 4-56 


10-176202! 19 

175928J i3 


42 


74417* 


3 


16 


920099 1 


40 


82^072 


4-56 


43 


74436i 


3 


10 


920015 1 


40 


824345 


4-56 


I75655| 17 


44 


74455o 


3 


i5 


919931 1 
919846 1 


41 


824619 


4-56 


1753811 16 


45 


744739 


3 


i5 


41 


824893 


4-56 


175107, i5 


46 


7449^3 


3 


i5 


919762 1 


41 


825i66 


4-56 


174334! '4 


47 


745ii7 


3 


i5 


919677 1 


41 


82543o 


4-55 


1 7456 1 i3 


48 


7i53o6 


3 


U 


919593 1 


41 


825713 


4-55 


I74287 12 


49 


7 15494 


3 


14 


919508 1 


41 


82 '986 


4-55 


174C14 11 


5o 


745683 


3 


'4 


919424 1 


41 


826259 


4-55 


173741 10 


5i 


9-745871 


3 


'4 


9-919339 1 


41 


9-826532 


4-55 


10-173468I 

i 7 3i 9 5| 8 
1729221 7 


52 


746o5o 
746248 


3 


14 


919254 1 


41 


8268o5 


4-55 


53 


3 


.3 


919169 1 


41 


827078 


4-55 


54 


746436 


4 


i3 


919080 1 


41 


827301 


4-55 


1726491 ■ 6 


55 


746624 


3 


i3 


919000 1 


41 


827624 


4-55 


172376 5 


56 


746812 


3 


i3 


918915 1 


42 


827897 


4-54 


I72io3| i 


57 


746999 
747187 


3 


i3 


9i883o'i 


42 


828170 


4-54 


171830; 3 


58 


3 


12 


9i8745,i 


42 


828442 


4-54 


I7l558 2 


5 9 


747374 


3- 


12 


918659)1 


42 


828715 


4-54 


I7I285 


1 


6o 


74756a 


3- 


12 


918574I1 


42 


828987 


4-54 


I7IOI3 





Cosine 





D. 


^ine |5 


6° 


Cotang. ! 


D 


Tang. 


JL. 



62 


(81 


t DEGREES ) A 


TABLE OF LOGARITHMIC 




M. 


Sine 


D. 


CoBino J D. 


Tang. 


D. 


Cotang. [ 





9-747562 


3-12 


9-9i8574!i-42 


9-828987 


4 


•54 


10-171013 60 1 


i 


747749 


3-12 


918489:1-42 


829260 


4 


•54 


170740 


% { 


2 


7479 36 


3-12 


918404 1 -42 


829532 


4 


•54 


17046S 


3 


748123 


3- II 


9i83i8|i-42 


829805 


4 


•54 


1 70 1 95 


57 


4 


7483io 


3-u 


9i8233[i-42 


830077 


4 


•54 


169923 


56 


5 


748497 


3-ii 


918147 


1-42 


83o349 


4 


•53 


i6 9 65 1 


55 


6 


748683 


3-u 


918062 


1-42 


83o62i 


4 


•53 


169379 


54 


7 


748870 


3-u 


917976 


1-43 


83o8 9 3 


4 


•53 


169107 


53 


8 


749056 


3-io 


917891 


1-43 


83n65 


4 


•53 


i 68835 


52 


9 


749243 


3- 10 


917805 


1-43 


831437 


4 


•53 


168563 


5i 


10 


749429 


3- 10 


917719 


i-43 


831709 


4 


•53 


168291 


5o 


ii 


9-749615 


3- 10 


9-91763411 -43 


9-831981 


4 


•53 


io- 168019 


% 


12 


749801 


3-io 


917548 i-43 


832253 


4 


•53 


167747 


i3 


749987 


3-09 


917462 i-43 


832525 


4 


•53 


167475 


47 


U 


750172 


3-09 


917376 1 -43 
917290 1 -43 


832796 


4 


•53 


167204 


46 


ID 


75o358 


3-09 


833o68 


4 


•52 


166932 


45 


16 


75o543 


3- 00 


91720411-43 


833339 


4 


•32 


1 6666 1 


44 


17 


750729 


3-09 


91 71 18I 1 -44 


8336n 


4 


52 


i6638 9 


43 


18 


750914 


3-o8 


917032 i-44 


833882 


4 


•32 


166118 


42 


*9 


751099 


3-o8 


916946 i-44 


834i54 


4 


52 


165846 


4i 


20 


7^1284 


3-o8 


9 i685 9 1-44 


834425 


4 


52 


165375 


40 


21 


9-751469 


3-o8 


9-916773 1-44 


9-834696 


4 


52 


io- i653o4 


3 9 


22 


75i654 


3-o8 


916687 


1-44 


804967 


4 


•32 


i65o33 


38 


23 


751839 


3- 08 


916600 


1 -44 


835233 


4 


32 


164762 


% 


24 


752D23 


3-07 


9i65i4 


1-44 


8355o9 


4 


32 


1 6449 1 


23 


752208 


3-07 


916427 


1-44 


835 7 3o 


4 


5 I 


164220 


35 


26 


752392 


3-07 


916341 


i-44 


836o5i 


4 


31 


163949 


34 


27 


752376 


3-07 


91625411-44 


836322 


4 


5l 


163678 


33 


28 


732760 


3-07 


916167 1 -45 


836393 


4 


5i 


163407 


32 


?9 


752944 


3-o6 


916081 1 .45 


836S64 


4 


31 


1 63 1 36 


3i 


3o 


733128 


3-o6 


915994 1-43 


83 7 i34 


4 


5i 


162866 


3o 


3.1 


9-7533i2 


3-o6 


9-915907 1-45 


9-8374o5 


4 


5i 


io- 162395 


3 


32 


753495 


3- 06 


915820 i-45 


837675 


4 


5.i 


162325 


33 


753679 


3-o6 


913733 


1-45 


83 79 46 


4 


5i 


162054 


27 


34 


7 53 86 2 


3-o5 


915646 


i-45 


8382i6 


4 


5t 


161 784 


26 


35 


754046 


3-o5 


915339 


1-43 


838487 


4 


30 


16 1 5i3 


25 


36 


754229 


3-o5 


915472 


1-43 


838 7 5 7 


4 


5o 


161 243 


24 


37 


734412 


3-o5 


915385 


1-45 


839027 


4 


30 


160973 


23 


38 


754595 


3-o5 


915297 


1-45 


83 9 2 97 


4 


5o 


160703 


22 


3 9 


754778 


3-o4 


915210 


1-45 


83 9 568 


4 


5o 


i6o432 


21 


40 


734960 


3-o4 


9i5i23 


1-46 


83g838 


4 


5o 


160162 


20 


41 


9-755i43 


3 • 04 


9"9i5o35 


1.46 


9-840108 


4 


30 


10-159892 


\l 


41 


755326 


3 -oi 


914948 


1-46 


840378 


4 


5o 


159622 


43 


7355o8 


3-o4 


9 1 4860 


1-46 


840647 


4 


5o 


i5 9 353 


17 


44 


753690 


3-o4 


91477 3 


1-46 


840917 


4 


49 


159083 

i588i3 


16 


45 


735872 


3-o3 


9U685 


1-46 


841 187 


4 


49 


i5 


46 


736054 


3-o3 


914598 


1-46 


84U57 


4 


49 


* i58543 


14 


47 


756236 


3-o3 


914510 


1-46 


841726 


4 


49 


158274 


i3 


48 


. 756418 


3-o3 


9'4422 


1-46 


841996 


4 


49 


i58oo4 


12 


49 


736600 


3-o3 


914334 


1-46 


842266 


4 


49 


137734 


11 


5o 


756782 


3-02 


914246 


1.47 


842 535 


4 


49 


157465 


10 


5 1 


9-756963 


3-02 


9-9UI58 


1-47 


9-842805 


4 


49 


io- 1 5719D 


I 


02 


757U4 


3-02 


914070 


1.47 


843074 


4 


49 


156926 


53 


757326 


3-02 


913982 


1.47 


843343 


4 


49 


156657 


1 


54 


757507 


3-02 


913894 


i-47 


8436.2 


4 


% 


156388 


6 


55 


757688 


3-ot 


913S06 


1-47 


843882 


4- 


i56n8 


5 


56 


737869 


3-oi 


913718 


1-47 


844i5i 


4 


48 


155849 4 


h 


758o5o 


3-oi 


9i363o 


1.47 


844420 


4 


4S 


1 5558o 


3 


58 


75823o 


3oi 


913541 


1-47 


844689 


4 


48 


i553u 


2 


5 9 


75841 1 


3oi 


9i3453 1 -47 


844958 


4 


48 


i55o42 


1 


60 


758591 


3-oi 


9 1 3365 1 -47 


845227 


4 


48 


154773 





L 


Cosine 


D. 


Sine 55° 


Cotang. 


D. 


Tang- 


M. 





s 


tNES AND TANGENTS. 


(35 DEGREES. 


) 


53 


*M. 


Sine 


D. 


Cosine 


D. 


Tang. 


D. 


1 Cotang. 




o 


9-758591 


3-oi 


9-9i3365 


1-47 


9-845227 


4-48 


10-154773 


60 


t 


738772 


3-oo 


913276 


i-47 


845496 


4-48 


1 545o4 


5 9 


2 


758952 


3-oo 


913187 


i-48 


845764 


4-48 


» 1 54236 


58 


3 


759132 


3-oo 


913099 


1.48 


846o33 


4-48 


153967 


57 


4 


P9312 


3-oo 


9i3oio 


1-48 


846302 


4-48 


1 536 9 8 


56 


5 


75949a 


3 -oo 


912922 


1.48 


8465 7 o 


4-47 


i5343o 


55 


6 


739672 


2-99 


912833 i-48 


846839 


4-47 


i53i6i 


54 


7 


75 9 852 


2-99 


912744 1 ■ 43 


84710- 


4-47 


152893 


53 


8 


76003 1 


2-99 


912655 1 -48 


847376 


4-47 


152624 


52 


9 


760211 


2-99 


912566 i-48 


847644 


4-47 


i5 2 356 


5i 


10 


760390 


2-99 
2- 98 


912477 U-48 


8479 ! 3 


4-47 


152087 


5o 


n 


9-760569 


9-912388 1-48 


9-848181 


4-47 


io-i5i8i9 


49 


12 


760748 


2- 9 8 


912299(1-49 


848449 


4-47 


i5i55i 


48 


i3 


760927 


2- 9 8 


912210 i-49 


848717 


4-47 


i5i283 


47 


U 


761 106 


2-98 


912121 


1-49 


848986 


4-47 


i5ioi4 


46 


i5 


761285 


2- 9 8 


QI2o3l 


i-4g 


849254 


4-47 


1 50746 


45 


16 


761464 


2- 9 3 


91 1942 1-49 


849522 


4-47 


150478 


44 


n 


761642 


2-97 


9iiS53 1 -49 


84979° 


4.46 


150210 


43 


iS 


761821 


2-97 


911763 1-49 


85oo58 


4-46 


149942 


42 


19 


761999 


2-97 


911674(1 -49 


85o325 


4-46 


149675 


4i 


20 


762177 


2-97 


9ii5S4ji -49 


85o5 9 3 


4-46 


149407 


40 


21 


9«762356 


2-97 


9-91 I4g5ji -49 


9-85o86l 


4-46 


10-149139 


39 


22 


762534 


2-96 


9H4o5 1-49 


831129 
85i3 9 6 


4-46 


1 4887 1 


38 


23 


762712 


2-96 


91 i3i5 1 -5o 


4-46 


148604 


37 


2 t 


762889 


2-96 


91 1 226' 1 -5o 


85 1 664 


4.46 


U8336 


36 


25 


763067 
763245 


2-96 


91 1 i36J 1 - 5o 


85i 9 3i 


4-46 


148069! 35 


26 


2-96 


91 1046 1 -5o 


852199 
832466 


4-46 


147801I 34 


27 


763422 


2-96 


910956! 1 -5o 


4.46 


147534! 33 


23 


763600 


2-g5 


910866 1 -5o 


852733 


4-45 


147267 32 


? 9 


7^777 


2-93 


910776! 1 -5o 
910686 i- 5o 


853ooi 


4-45 


146999: 3i 


3o 


7639^4 


2- 9 5 


853268 


4-45 


146732! 3o 


? [ 


9 -764131 


2- 9 5 


9-9105961 -5o 


9-853535 


4-45 


10 - 146465 29 


32 


764308 


2-93 


9io5o6 1 -5o 


8538o2 


4-45 


146198! 28 


33 


764485 


2-94 


9io4i5,J -5o 


854069 


4-45 


14593 1! 27 


34 


764662 


2-94 


9io325,i -5i 


854336 


4-45 


' 145664: 26 


35 


764838 


2-94 


910235 1 -5i 


8546o3 


4-45 


145397! 25 


36 


765oi 5 


2- 9 4 


9ioi44ri -5i 


854870 


4-45 


i45i3o! 24 


3 7 


765191 


2-94 


910054 1 -5i 


855i3 7 


4-45 


1 44863| 23 


3S 


765367 


2- 9 4 


909963 ! 1 • 5 1 


855404 


4-45 


144596 


22 


3 9 


765544 


2-93 


909873,1-51 


8556 7 r 


4-44 


U4329 


21 


4o 


760720 


2-93 


9097(82 i- 5 1 


855 9 38 


4-44 


144062 


20 


4 1 


9-765896 


2-93 


9-909691 1 -5i 


9*856204 


4-44 


10-143796 


19 


42 


766072 


2-93 


909601 1 -5i 


856471 


4-44 


U3529 


18 


43 


766247 


2- 9 3 


909510. i«- 5 1 


856737 


4.44 


U3263 


17 


44 


766423 


2-93 


909419'! -5i 
909328 I -52 


857004 


4-44 


142996 


16 


45 


766598 


2-92 


857270 


4-44 


142730 


i5 


46 


766774 


2-92 


909237*1 -52 


857537 


4-44 


142463 


14 


47 


766949 


2-Q2 


909146 1 -52 


837803 


4-44 


U2197 


i3 


48 


767124 


2-92 


909055' I -52 


858o6 Q 


4-44 


141931 


12 


49 


767300 


2-92 


908964' 1-52 


853336 


4.44 


1 41 664 


11 


5o 


767475 


2-91 


908873I I -52 


853602 


4-43 


I4i3 9 8 


10 


5i 


9.767649 


2-9I 


9-908781 1-52 


9-858868 


4-43 


io-i4ii32 (f 


52 


767824 


2-91 


908690 I -52 


859134 


4-43 


140866 


8 


53 


767999 


2-9I 


908599^1 -52 


859400 


4-43 


1 40600 


7 


5 4 


768173 


2-9I 


908507I1 -52 


35 9 666 


4-43 


!4o334 


6 


55 


766348 


2-90 


908416,1-53 


85 99 32 


4-43 


140068 


5 


56 


768322 


2-90 


908324; 1 -53 


860198 


4-43 


139802 


4 


£ 7 


768697 


2-90 


9o8233;i-53 


860464 


4-43 


139336 


3 


58 


768871' 


2-90 


90814111 -53 


860730 


4-43 


139270 


2 


5 9 


769045 ; 


2-90 


908049 1 -53 


860995 


4-43 ' 


1 39005 1 


6o 


7692I9 


2-90 


907958k -53 


861 261 


4 43 ! 


i38 7 3 9 




Cosine 


~TX~\ 


bine 151° 


Cotang" 


~D7-\ 


Tang. ! 


M. j 



64 


(3G 

Sine 


DEGREES.) A 


TABLE OF LOGARITHMIC 




rr 


D. 


Cosine | D. 


Tang. 


D. 


Cotaug, 


< 
60 


, o 


9.769219 
769393 


2-90 


9-90795811 '53 


9-861261 


4.43 


10.138739 
13847.3 


I 


2 


S 


90786611 '53 


8O1527 


4 


43 


u 


2 


769566 


2 


89 


907774! 1 


53 


861792 


4 


42 


138208 


3 


769740 


2 


89 


907682,1 


53 


862058 


4 


42 


137942 


57 


4 


769913 


2 


89 


907590 1 


53 


862323 


4 


42 


137677 


56 


5 


770087 


2 


89 


907498:1 


53 


862589 


4 


42 


1 3741 1 


55 


6 


770260 


2 


88 


907406 1 


53 


862354 


4 


42 


.37146 


54 


7 


770433 


2 


88 


9073 14' 1 


54 


863 1 19 

863385 


4 


42 


13688 1 


53 


8 


770606 


2 


88 


907-222 i 


54 


4 


42 


i366i5 


52 


9 


770779 


2 


88 


907 1 29' 1 


54 


86365o 


4 


42 


i3635o 


5i 


10 


770952 


2 


83 


907087 1 


54 


863gi 5 


4 


42 


i36o85 


5o 


1 1 


9-771125 


2 


88 


9-906945 1 


54 


9-864180 


4 


42 


io- 135820 


49 


12 


771298 


2 


37 


9o6852 : 1 


54 


364445 


4 


42 


1355.55 


48 


i3 


77 1 470 


2 


87 


906760 I 


54 


864710 


4 


42 


135290 


47 


1 4 


771643 


2 


87 


906667 , 1 


54 


864975 


4 


41 


i35o25 


46 


i5 


77 i8i5 


2 


37 


9065751 


54 


865240 


4 


4« 


134760 


45 


16 


7719^7 


2 


87 


906482 1 


54 


8655o5 


4 


4i 


134495 


44 


n 


772159 


2 


87 


9o638 9( i 


55 


865 77 o 


4 


4i 


i3423o 


43 


18 


772331 


2 


86 


906296,1 


55 


866o35 


4 


4i 


133965 


42 


19 


7725o3 


2 


86 


900204,1 


55 


8663oo 


4 


4i 


133700 


41 


20 


772675 


2 


86 


9061 1 1 




55 


866564 


4 


4i 


133436 


40 


21 


9-772847 
773018 


2 


86 


9-906018 




55 


9-866829 


4 


4i 


to- 133171 


3 9 


22 


2 


86 


905925 


1 


55 


867094 


4 


4i 


132906 


38 


23 


773190 


2 


86 


9o5S32 




55 


867358 


4 


4i 


132642 


37 


24 


77336i 


2 


85 


905739 
9o5645 


1 


55 


867623 


4 


4i 


il2377 


36 


25 


773533 


2 


85 




55 


867887 


4 


4i 


i32ii3 


35 


26 


773704 


2 


85 


9o5552 




55 


863i52 


4 


4o 


i3i848 


34 


27 


77 38 7 5 


2 


83 


905459 


j 


55 


868416 


4 


4o 


i3i584 


33 


28 


774046 


2 


35 


9o5366 




56 


868680 


4 


4o 


i3 i32o 


32 


29 


7742H 


2 


85 


905272 




56 


868945 


4 


4o 


i3io55 


3i 


3o 


774338 


2 


84 


905179 




56 


869209 


4 


4o 


1 30794 


3o 


3i 


9-774558 


2 


8i 


y-9o5o85 




56 


9-869473 


4 


4o 


io- i3o527 


2 2 


32 


774729 


2 




904992 




56 


869737 


4 


4o 


i3o263 


28 


33 


774399 


2 


84 


904898 




56 


870001 


4 


4o 


1 29999 


27 


34 


775070 


2 


84 


904804 




56 


870265 


4 


40 


129735 


26 


35 


775240 


2 


84 


9347 1 1 

904617 




56 


870529 


4 


4o 


129471 


25 


36 


7754io 


2 


83 




56 


870793 
871037 


4 


40 


129207 


24 


37 


775j8o 


? 


83 


904523 




56 


4 


40 


1 28943 


23 


2 8 


77^750 


2 


83 


904429 




5 7 


8 7 i32i 


4 


40 


12S670 


22 


h 


775920 


2 


83 


904335 




5 7 


87i535 


4 


40 


128415 


21 


40 


776000 


2 


83 


904241 




r »7 


871849 


4 


3 9 


i28i5i 


20 


4* 


9-776259 


2 


83 


9-904147 




57 


9-872112 


4 


3 9 


10-127888 


13 


42 


77 6 42o 
776098 


2 


?i 


904053 




57 


872376 


4 


3 9 


127624 


43 


2 


82 


903959 




5 7 


872640 


4 


3 9 


127360 


17 


44 


776768 


2 


82 


903364 




57 


872903 


4 


39 


127097 


16 


45 


776937 


2 


82 


903770 




5 7 


8 7 3i6 7 


4 


3? 


1 26833 


[5 


46 


777106 


2 


82 


903676 




57 


873430 


4 


39 


1 26570 


i4 


3 


777275 


2 


81 


9o358i 


, 


57 


873694 
873957 


4 


3 9 


1 263o6 


i3 


777444 


2 


81 


903487 




57 


4 


39 


! 26043 


12 


49 


777613 


2 


Si 


903392 




58 


874220 


4 


39 


125780 


1 1 


5o 


777781 


2 


81 


903298 




58 


8744S4 


4 


39 


I255i6 


10 


5i 


9-777950 


2 


8! 


9-9o32o3 




58 


9.874747 


4 


39 


10 i?5253 


I 


52 


778119 


2 


81 


903 1 08 




58 


875010 


4 


39 


1 24990 


f3 


778287 


2 


80 


9o3oi4 


, 


58 


875273 


4 


33 


124727 


7 


64 


778455 


2 


80 


902019 
902824 


, 


58 


8 7 5536 


4 


38 


1 24464 


6 


55 


778624 


2 


80 




58 


875800 


4 


33 


124200 


5 


fo 


778792 


2 


80 


902729 


I 


58 


876063 


4 


38 


123937 


4 


^7 


778960 


2 


80 


902634 


1 


58 


876326 


4 


38 


.28674 


3 


58 


779128 


2 


80 . 


902539 




5 9 


876589 


4 


38 


1 23411 


2 


5 9 


779295 


2 


79 


902444 


1 


5 9 


8 7 685i 


4 


38 


128 1 491 1 


60 


779463 


2.79 


902349 

Sine 


1 


5g 


877114 
Totalis:. 


4 


38 


122886] it 


L 


Cosine 


I\ 


5 


3° 


D. 


tans:. 


M. 





SIXES AN'D TANGENTS. 


(37 DEGREES. 


) 


55 


M. 
o 


Sine 


1 D - 


Cosine. | D. 


I Tung. 


D. 


Cotj_ng. 


r 


9-779463 


2-79 


9-902349 1 09! 9-877114 


4-38 


10-122886 


\6o 


i 


779631 


2-79 


902253 1 -5g 


i 877377 


4-38 


122623 


\% 


2 


77979 s 


2-79 


902 1 58 1 • 5g 


1 877640 


4-38 


122360 


3 


779966 


2' 79 


902063 i-5cj 


877903 


4-38 


122097 


57 


4 


7 8oi33 


2-79 


901967 1 • 5g 


8 7 8i65 


4-38 


I2i835 


56 


5 


78o3oo 


2-78 


901872 1 -5g 


878428 


4-38 


121572 


55 


6 


780467 


2-78 


901776,1 .59 


878691 


4-38 


121309 


54 


7 


780634 


2-78 


901681 |i -59 


878953 


4-37 


121047 


53 


8 


780801 


2-78 


901085.1.59 


879216 


4-37 


120784 


52 


9 


780968 


2- 7 3 


901490; 1 -59 


879478 


4-3 7 


120522 


5i 


10 


781134 


2-78 


901394 1 -6o 


879"4i 


4-3 7 


120259 


5o 


ii 


9-781301 


2-77 


9-901298 1 -6o 


9-88uoo3 


4-37 


*o- 1 19997 


% 


12 


781468 


2-77 


901202^ -6o 


88o265 


4-37 


1 19735 


i3 


182634 


2-77 


90iio6;i-6o 


88o528 


4-3 7 


1 19472 


47 


i4 


781800 


2-77 


90ioiO|i -6o 


{280790 


4-3 7 


1 192 10 

118948 


46 


l5 


781966 


2-77 


900914 1 -6o 


88io52 


4-3 7 


45 


16 


782132 


2-77 


90o8i8|i-6o 


88i3i4 


4-37 


1 18686 


44 


17 


782298 


2-76 


900722J -6o 


88i5 7 6 


4-3 7 


1 1 8424 


43 


18 


7S2464 


2-76 


900626 1 -6o 


88i83o 


4-3 7 


118161 


42 


*9 


782630 


2-76 


90o529 ! i-6c 
900433 1 -6i 


882 1 01 


4-3 7 


1 17899 


4i 


20 


782796 


2-76 


882363 


4-36 


1 17637 


40 


21 


9.782961 


2-76 


9-90033711-61 


9.882625 


4-36 


io- 1 17375 


3 9 


22 


783127 


2-76 


9002401 1 '61 


882887 
883 148 


' 4-36 


117113 


38 


23 


783292 


2- 7 5 


90oi44>i -6i 


4-36 


n6S52 


37 


24 


733458 


2.75 


000047 J i-6 1 


883410 


4-36 


1 16590 


36 


25 


7 83623 


2.75 


899951 11-61 
8998041-61 


883672 


4-36 


1 16328 


35 


26 


783788 


2-70 


883 9 34 


4-35 


1 16066 


34 


2 


7 83 9 53 


2.75 


899707 1 .61 


884196 
884457 


4-36 


n58o4 


33 


784118 


2-70 


899660 1 -6i 


4-36 


1 i5543 


32 


29 


784282 


2-74 


899564 i- 61 


884719 


4-36 


ii528i 


3i 


3o 


784447 


2-74 


899467 1-62 


884980 


4 = 36 


Il5020 


3o 


3i 


9-784612 


2.74 


9-899370 1.62 


9-885242 


4-36 


10-114758 


3 


32 


784776 


2-74 


899273 1-62 


8855o3 


4-36 


I 14497 


33 


784941 


2-74 


89917611 .62 


835 7 65 


4-36 


114235 


27 


34 


785 1 o5 


2-74 


899078 
898981 


1-62 


8S6026 


4-36 


1 13974 


26 


35 


780269 


2. 7 3 


1 -62 


8S6288 


4-36 


113712 


25 


36 


785433 


2-73 


898884 


1-62 


8S6549 


4-35 


ii345i 


24 


u 


785597 


2-73 


898787 


1 -62 


886810 


4-35 


113190 


23 


785761 


2- 7 3 


898689' 1-62 


887072 


4-35 


1 12928 


22 


39 


785925 


S.73 


898092 1 -62 


8S7333 


4-35 


112667 


21 


40 


786089 


2.73 


8984941 1 -63 


88 7 5 9 4 


4-35 


1 1 2406 


20 


41 


9-786252 


2- 7 2 


9-898397 


i-63 


9.887855 


4-35 


10-112145 


\l 


42 


786416 


2-72 


898299 


1-63 


8881 16 


4-35 


1 1 1884 


43 


786579 


2-72 


898202 


1-63 


888377 


4-35 


111623 


17 


44 


786742 


2-72 


89S104 


i-63 


88863 9 


4-«5 


iii36i 


16 


45 


786906 


2- 7 2 


898006 


i-63 


888900 


4-35 


IIIIOO 


i5 


46 


787069 


2-72 


897908 


1-63 


889160 


4-35 


1 1 0840 


14 


47 


787232 


2-71 


897810 


i-63 


889421 


4-35 


1 10579 i3 


48 


787395 


2- 7 I 


897712 


1-63 


889682 


4-35 


iio3i8 12 


49 


787307 


2. 7 t 


897614 


1-63 


889943 


4-35 


noo57j 11 


5o 


787720 


2-71 


897516 


i-63 


890204 


4-34 


109796; 10 


5i 


9.787883 


2-71 


9-897418 


1-64 


9-890465 


4-34 


10-109535 9 
109275 8 


52 


788045 


2.71 


897320 


1-64 


890725 


4-34 


53 


788208 


2-71 


897222 


1.64 


890986 


4-34 


109014 7 ■ 
108753 6 


54 


788370 


2-70 


897123 


1-64 


891247 


4-34 


55 


788532 


2- tO 


897025 


1-64 


891507 


4.34 


io84o3 
1082J2 


5 


56 


788694 


2-70 


896926 


1-64 


891768 


4.34 


4 


s 


788856 


2-70 


896828 


1-64 


892028 


4.34 


107972 


3 


' 789018 


2-70 


896729 


1-64 


892289 


4-34 


■ 107711 


2 


59 


789180 


2-70 


8 9 663 1 


1-64 


892549 


4-34 


107451 


1 


60 


789342 


2-69 


896532 


1-64 


892810 


4.34 


107190 







Cosire 


D. 


Sine 


52° 


Cotansr. 1 


D. 


Tang. 


M. 



56 


(- 


3 DEGREES.) A 


TABLE OF L< 


3GARITHMIC 




M. 


Sine 


D. 


Cosine 


D. 


Tang. 


I). 


Cctang. 




o 


9.789342 


2-69 


9-896532 


1-64 


9-892810 


4-34 


io- 107190 


60 


i 


789504 


2-69 


896433 


1 -65 


893070 


4.34 


106930 


5 2 

58 


2 


789665 


2-69 


8 9 6335 


i-65 


8 9 333 1 


4-34 


1 06669 


3 


7S9827 


2-69 


896236 


1 -65 


893591 
8 9 3S5i 


4-34 


106409 


57 


4 


7S99S8 


2-69 


896137 


1-65 


4-34 


106149 


56 


5 


79°'49 


2-69 


8 9 6o38 


i-65 


8941 11 


4-34 


105889 


55 


6 


7903 10 


2-68 


895939 
895840 


i-65 


894371 


4-34 


105629 


54 


I 


790471 


2-68 


1-65 


894632 


4-33 


io5368 


53 


790632 


2-68 


895741 


1 -65 


894892 


4-33 


io5io8 


52 


9 


790793 


2-68 


8 9 564i 


i-65 


8 9 5i52 


4-33 


104848 


5i 


10 


790934 


2-68 


895542 


i-65 


895412 


4-33 


104588 


5o 


1 1 


9-79UI5 


2-68 


9-895443 


1-66 


9-893672 


4-33 


10-10432? 


% 


12 


791275 


2-67 


895343 


i-66 


893932 


4-33 


1 04068 


i3 


79U36 


2-67 


895244 


1.66 


896192 


4-33 


io38o8 


47 


i4 


791 5o6 
791757 


2-67 


895145 


1.66 


896452 


4-33 


io3548 


46 


i5 


2«67 


8 9 5o45 


1.66 


896712 


4-33 


103288 


45 


16 


791917 


2-67 


894945 
894846 


1.66 


896971 


4-33 


103029 


44 


17 


792077 


2-67 


1-66 


897231 


4-33 


102769 


43 


18 


792237 


2-66 


894746 


1-66 


897491 
897751 


4-33 


102509 


42 


i9 


792397 
792537 


2-66 


894646 


1.66 


4-33 


102249 


41 


20 


2-66 


894546 


i-66 


898010 


4-33 


101990 
10-101730 


40 


21 


9-792716 


2-66 


9-894446 


1-67 


9-898270 


4-33 


ll 


22 


792876 


2-66 


894346 


1.67 


8 9 853o 


4-33 


101470 


23 


7q3o35 


2-66 


894246 


1.67 


898789 


4-33 


101211 


37 


24 


793i ? 5 


2-65 


894146 


1.67 


899049 
899308 


4-32 


1 0095 1 


36 


25 


7 9 33o4 


2-65 


894046 


1.67 


4-32 


100692 


35 


26 


7935i4 


2-65 


8 9 3 9 46 
893846 


i -67 


89956S 


4-32 ■ 


100432 


34 


27 


793673 


2-65 


1-67 


899827 


4-32 


100173 


33 


28 


793832 


2-65 


893745 


1.67 


900086 


4-32 


099914 


32 


I 9 


793991 


2-65 


8 9 3645 


1.67 


900346 


4-32 


099654 


3i 


3o 


794i5o 


2-64 


893544 


1.67 


900605 


4-32 


099395 

10-099136 

098876 


3o 


3i 


9 -7943o8 


2-64 


9 -8 9 3444 


i-68 


9-900864 


4-32 


29 


32 


794467 


2-64 


893343 


i-68 


901124 


4-32 


28 


33 


794626 


2-64 


893243 


i-68 


9 oi383 


4-32 


098617 


11 


34 


794734 


2-64 


8 9 3i42 


i-68 


901642 


4-32 


098358 


35 


794942 


2-64 


8g3o4i 


i-68 


901901 


4-32 


098099 


25 


36 


793101 


2-64 


892940 


i-68 


902160 


4-32 


097840 


24 


37 


7 9 525 9 


2-63 


892839 


i-68 


902419 


4-32 


097581 


23 


38 


795417 


2-63 


8 9 2 7 3 9 


i-68 


902670 
902938 


4-32 


097321 


22 


3 9 


795375 


2-63 


8 9 2638 


i-68 


4-32 


097062 


21 


40 


795733 


2-63 


8 9 2536 


i-68 


903197 


4-3i 


096803 


20 


4i 


9 -7 9 58 9 i 


2-63 


9.892435 


1 -69 


9-903455 


4-3i 


10-096545 


\t 


42 


796049 


2-63 


892334 


1 -69 


903714 


4-3i 


096286 


43 


796206 


2-63 


892233 


1 -69 


903973 


4-3i 


096027 


n 


44 


796364 


2-62 


892132 


1-69 


904232 


4-3i 


095768 


16 


45 


796521 


2-62 


892030 


1 -69 


904491 
904730 


4-3i 


095509 


i5 


46 


796679 


2-62 


891929 
891827 


1-69 


4-3r 


095250 


14 


% 


796836 


2-62 


1 -6g 


905008 


4-3i 


094992 
094703 


i3 


796993 


2-62 


891726 


1-69 


905267 


4-3i 


12 


49 


797i5o 


2 -61 


891624 


1-69 


905526 


4-3i 


094474 


11 


5o 


797307 


2-6l 


891523 


1.70 


905784 


4-3i 


094216 


10 


5i 


9-797464 


2-6l 


9-891421 


1-70 


9-906043 


4-3i 


10-093957 


9 


52 


797621 


2-6l 


891319 


1-70 


906302 


4-3i 


093698 


8 


53 


797777 


2-6l 


891217 


[•70 


906560 


4-3i 


093440 


7 


54 


797934 


2-6l 


891 1 i5 


[•70 


90681Q 


4-3i 


093181 


6 


55 


798091 


2- 6l 


891013 


I -70 


907077 


4«3i 


092923 


5 


56 


798247 


2-6l 


89091 1 
890809 


[•70 


907336 


4-3i 


092664 


4 


a 


798403 


2- 60 


[•70 


907594 


4-3i 


092406 


3 


798560 


2- 60 


890707 


[.70 


907852 
-908 1 1 1 


4-3i 


092148 


2 


5 9 


798716 


2- 60 


890605 


[•70 


4-3o 


091889 


1 


60 


798872 


2-6o 


8go5o3 


[ -70 


908369 


4-3o 


091631 







Cosine 


D. 


Sine 


51° 


Cotan<?~ 


™~d; 


Tang. 


M. 





SINKS 


AND TANGENTS. 


(39 LEUREE3. 1 


> 


57 


M. 
o 


Sine 


D. 


Cosine | D. 


Tang. 


D. 


Cotang. 




0-798872 


2 


•60 


9- 890503 1 -70 


9-908369 

908628 


4-3o 


10-091631 


60 


i 


799028 


2 


• 6o 


890400 1 1 


•7i 


4-3o 


091372 


ft 


2 


799184 


2 


•60 


890298^1 


•7' 


908886 


4-3o 


091 1 14 


3 


79 9 33 9 


2 


•5 9 


890195 1 


•7i 


909144 


4-3o 


090856 


57 


4 


79 9 4o5 
79965 1 


2 


5 9 


890093 1 


•7' 


909402 


4-3o 


090598 56 


5 


2 


5 9 


8099901 1 


•v 


909660 


4-3o 


090340 


55 


6 


799806 


2 


.59 


889888 1 


•7i 


909918 


4-3o 


090082 
089823 


54 


I 


799962 
8001 17 


2 


5 9 


88 97 85 ' 1 


•7' 


910177 


4-3o 


53 


2 


5 9 


889682 1 
8895791 


V 


910435 


4-3o 


o8 9 565 


52 


9 


800272 


2 


58 


V 


910693 


4-3o 


089307 


5i 


IO 


800427 


2 


58 


889477! 1 


•7i 


" 910901 


4-3o 


0S9049 


5o 


II 


9- 8oo582 


2 


58 


9.889374I1 


72 


9-911209 


4-3o 


10-088791 


% 


12 


800737 


2 


58 


889271 1 


72 


91 1467 


4-3o 


088533 


i3 


800892 


2 


58 


889168 1 


•72 


911724 


4-3o 


088276 


47 


14 


80 r 047 


2 


53 


8890641 


•72 


911982 


4-3o 


088c 1 8 


46 


i5 


801201 


2 


58 


888961 1 


■72 


912240 


4-3o 


087760 


45 


16 


8oi356 


2 


57 


888858 1 


72 


912498 
9127D6 


4-3o 


087502 


44 


«7 


8oi5n 


2 


57 


888755 1 


■72 


4-3o 


087244 


43 


18 


8oi665 


2 


5 7 


88865i 1 1 


72 


9i3oi4 


4-29 


086986 


42 


>9 


801819 


2 


5 7 


#88548j 1 


72 


913271 


4-29 


086729 


4i 


20 


801973 


2 


5 7 


8884441 


73 


913529 


4-29 


086471 


40 


21 


9-802128 


2 


57 


9-888341 1 


73 


9-913787 


4-29 


io-o862i3 


\l 


22 


802282 


2 


56 


888237 1 


73 


914044 


4-29 


o85 9 56 


23 


802436 


2 


56 


888i34 1 


73 


9U3o2 


4-29 


085698 


3? 


24 


802589 
802743 


2 


56 


888o3o 




73 


9i456o 


4-29 


085440 


36 


25 


2 


56 


887926 




73 


914817 


4-29 


o85i83 


35 


26 


802897 
8o3o5o 


2 


56 


887822 




73 


915075 


4'2Q 


084925 


34 


2 7 


2 


56 


887718 




/3 


9i5332 


4-29 


084668 


33 


28 


803204 


2 


56 


887614 




73 


915590 


4-29 


084410 


32 


2 9 


8o3357 


2 


55 


887510 




73 


915847 


4-29 


0841 53 


3i 


3o 


8o35u 


2 


55 


887406 




74 


916104 


4-29 


o838 9 6 
io- o83638 


3o 


3i 


9«8o3664 


2 


55- 


9-887302 




7-t 


9-916362 


4-29 


3 


32 


803817 


2 


55 


887198 




74 


916619 


4-29 


o8338i 


33 


803970 


2 


55 


887098 
886989 
886885 




74 


916877 


4-29 


o83i23 


27 


34 


804123 


2 


55 




74 


917^4 


4-29 


082866 


26 


35 


804276 


2 


54 




74 


917391 


4-29 


082609 


25 


36 


804428 


2 


54 


886780 




74 


917648 


4-29 


082352 


24 


2? 


8o458i 


2 


54 


886676 




74 


917905 


4-29 


082095 


23 


38 


804734 


2 


54 


8865 7 i 




74 


918163 


4-28 


081837 


22 


3 9 


804886 


2 


54 


886466 1 1 


74 


918420 


4-28 


o8i58o 


21 


4o 


8o5o39 


2 


54 


886362 1 


V 


918677 


4-28 


081 323 


20 


41 


9-805191 


2 


54 


9.886257 1 


75 


9'9i8934 


4-28 


10-081066 


\% 


42 


8o5343 


2 


53 


886i5 2 1 


V 


919191 


4-28 


080809 


43 


8o5495 


2 


53 


886047! 1 


7 J 


919448 


4-28 


o8o552 


17 


44 


8o5647 


2 


53 


885 9 42 
885837 




75 


919705 


4-28 


080295 16 


45 


805799 


2 


53 




V 


919962 


4-28 


o8oo38 1 5 


46 


8o595i 


2 


53 


885 7 32 




V 


920219 


4-28 


079781 


14 


47 


806 1 o3 


2 


53 


885627 




75 


920476 


4-28 


079524 


i3 


48 


806254 


2 


53 


885522 




75 


920733 


4-28 


079267 


12 


4o 


806406 


2 


52 


885416 




7 : » 


920990 


4-28 


079010 
078753 


11 


5o 


806557 


2 


52 


8853 1 1 




76 


921247 


4-28 


10 


5i 


9 • 806709 


2 


52 


9-885205 




76 


9-92i5o3 


4-28 


10-078497 


I 


52 


806860 


2 


52 


885 1 00 




76 


921760 


4-28 


078240 


P 


80701 1 


2 


52 


884994 




76 


922017 


4-28 


077983 


1 


5 4 


807163 


2 


52 


884889 




76 


922274 


4-28 


077726 


6 


55 


807314 


2 


52 


884783 




76 


922530 


4-28 


077470 


5 


56 


807465 


2 


5i 


884677 




76 


922787 


4-28 


077213 


4 


u 


8o 7 6i5 


2 


5i 


884572 




76 


923044 


4-28 


076956 


3 


807766 


2 


5i 


.884466 




76 


9233oo 


4-28 


076700 


2 


5 9 


807917 
808067 


2 


5i 


88436o 




76 


923557 


4-27 


076443 


I 


6o 


2- 


5i 


884254 




77 


9238i3 


4-27 
D- i 


076187 







Cosine 


D. 


Sine 


5p° 


Cotansr. 


Tang. 


_Mi, 



58 


(40 DEGREES.) A 


TABLE OF LOGARITHMIC 




M. 


Sine 


D. 


Cosine | I). 


Tang. 


I). 


1 Ootang. 


, 





q- 808067 


2-5l 


9-884254 


1.77 


9-9238i3 


4-27 


110-076187 


60 


i 


808218 


2-5l 


884148 


1.77 


924070 


4-27 


07593c 


5o 

58 


2 


8o8368 


2-5l 


884042 


1.77 


924327 


4-27 


075673 


3 


8o85i 9 


2-5o 


883 9 36 
883829 


'•77 


924583 


4-27 


075417 


57 


4 


8o366 9 


2-5o 


1.77 


924840 


4-2 7 


07516c 


56 


5 


808819 


2-5o 


883 7 23 


1.77 


925096 
925352 


4-27 


074904 


55 


6 


808969 


2-5o 


.883617 


1.77 


4- 27 


074648 


54 


7 


8091 19 


2-5o 


8835io 


1.77 


925609 
925865 


4-2 7 


074391 


53 


8 


809269 


2-5o 


883404 


1.77 


4-27 


074135 


5a 


9 


809419 


2-49 


883297 


1.78 


926122 


4-27 


073878 


5i 


10 


809369 

9-809718 


2-49 


• 883ioi 


1.78 


926378 


4-2 7 


073622 


5o 


1 1 


2-49 


9-883o84 


1.78 


9-926634 


4-27 


10-073366 


4 9 

48 


12 


809868 


2-49 


882077 
882871 


1.78 


926890 


4-2 7 


073110 


i3 


810017 


2-49 


1.78 


927147 


4-27 


072853 


47 


i4 


810167 


2-49 
2.48 


882764 


1.78 


927403 


4-2 7 


072597 


46 


*5 


8io3i6 


8S2657 


1.78 


927609 
927915 


4-2 7 


072341 


45 


■6 


8io465 


2.48 


88255o 


1.78 


4-27 


072085 


44 


«7 


810614 


2.48 


8S2443 


1.78 


928171 


4-27 


071829 


43 


18 


810763 


• 2-48 


882336 


1.79 


928427 


4-27 


07i5 7 3 


42 


»9 


810912 


2-48 


882229 


1.79 


92S683 


4-2 7 


071317 


4i 


20 


811061 


2-48 


882121 


1.79 


928940 


4-27 


071060 


40 


21 


9-811210 


2-48 


9-882014 


1.79 


9.929196 
929452 


4-27 


10 070804 


IS 


22 


8n358 


2-47 


881907 


1.79 


4-27. 


070548 


23 


8u5o7 


2-47 


8S1799 


1.79 


92970S 


4-2 7 


070292 


37 


24 


8n655 


2-47 


881692 
88 1 584 


1.79 


929964 


4-26 


070036 


36 


25 


81 1804 


2-47 


1.79 


930220 


4-26 


069780 


35 


26 


81 1952 


2-47 


881477 


1.79 


930475 


4-26 


069525 


34 


27 


812100 


2-47 


88i36o 


1.79 


930731 


4-26 


069269 


33 


28 


812248 


2-47 


881261 


1. 80 


930987 


4-26 


0690 1 j 

068757 


32 


29 


812396 


2-46 


88u53 


1.80 


93i243 


4-26 


3i 


3o 


8i2544 


2-46 


881046 


1.80 


931499 
9-931755 


4-26 


o085oi 


3o 


3i 


9-812692 


2-46 


9-88o 9 38 


1. 80 


4-26 


10-068245 


3 


32 


812840 


2-46 


88o83o 


1. 80 


932010 


4-26 


067990 


33 


812988 


2-46 


8S0722 


1. 80 


932266 


4-26 


0677J4 


ll 


34 


8i3i35 


2-46 


880613 


1. 80 


932522 


4-26 


067478 


35 


8i3283 


2-46 


88o5o5 


1. 80 


932778 


4-26 


067222 


25 


36 


8i343o 


2-45 


880397 


1.80 


933o33 


4-26 


066967 


24 


ll 


813578 


2.45 


880289 


1. 81 


933289 


4-26 


0667 1 1 


23 


8x3725 


2-45 


880180 


1. 81 


933545 


4-26 


o66455 


22 


3 9 


8i38 7 2 


2-45 


880072 


1. 81 


9338oo 


4-26 


' 066200 


21 


40 


814019 


2-45 


879963 


1. 81 


934o56 


4-26 


065944 


20 


41 


9-814166 


2-45 


9-8 79 855 


1. 81 


9-93431 1 


4-26 


io-o6568 9 


IS 


42 


8i43i3 


2-45 


879746 


1. 81 


934567 


4-26 


065433 


43 


814460 


2-44 


8 79 63 7 


1. 81 


934823 


4-26 


o65i77 


\l 


44 


814607 


2-44 


8 79 52 9 


1. 81 


935078 


4-26 


064922 


45 


8i4753 


2-44 


879420 


1. 81 


935333 


4-26 


064667 


i5 


46 


814900 


2-44 


8793 1 1 


1. 81 


9 3558 9 


4-26 


06441 1 


14 


47 


8 1 5o46 


2-44 


879202 


1-82 


935844 


4-26 


064 1 56 


i3 


48 


8i5i 9 3 


2-44 


879093 


1.82 


936100 


4-26 


063900 


12 


49 


8i533g' 


2-44 


878084 


1.82 


9 36355 


4-26 


o63645 


11 


5o 


8 1 5485 


2-43 


878875 


1.82 


936610 


4-26 


063390 


10 


5 1 


9-8i563i 


2-43 


9-878766 


1.82 


9-936866 


4-25 


io-o63i34 


9 


52 


815778 


2-43 


878656 


1.82 


937121 


4'2D 


062879 


8 


53 


815924 


2-43 


87S547 


1-82 


937376 


4-25 


062624 


7 


54 


816069 


2-43 


878438 


1.82 


937632 


4-25 


o62368 6 J 


55 


8i62i5 


2-43 


878328 


[.82 


937887 


4-25 


0621 13 


5 


56 


8i636i 


2-43 


878219 


[•83 


938142 


4-25 


o6i858 


4 


s 


816507 


2-42 


878109 


i-83 


938398 
9 3S653 


4-25 


061602 


3 


8i6652 


2-42 


877099 


r.83 


4-25 


061347 


2 


59 


816798 


2-42 


877800 
8777S0 


[-83 


938908 


4-25 


061092 
060837 


1 


60 


816943 


2-42 


[-83 


939163 


4-25 







Cosine 


D. 


Sine 


49° 


Cotang. 


D. 


Tang. 


m7 





S7WES A^D TATSTOENTS. 


(41 DKGRETBS. 


r 


S9 


c 


Sine 
0-81694') 


T>. 


Cosine 1 L>. 


Tung. 


D. 


Cotang. 




2-42 


9.877780 


1-83 


9-939163 


> 4-?5 


10-060837 


60 


I 


817088 


2-42 


877670 


i-83 


939418 


4-25 


o6o582 


5 9 


2 


817233 


2-42 


877560 


i-83 


939673 


4-23. 


060327 


58 


3 


817379 


2-42 


877450 


i-83 


939928 


4-25 


060072 


57 


4 


817524 


2-41 


877340 


1 83 


940183 


4-25 


059817 


56 


5 


817668 


2-41 


877230 


1-84 


940438 


4-25 


* 039562 


55 


6 


817813 


2-41 


877120 


1 84 


940694 


4-23 


059306 


54 


7 


817958 


2-41 


877010 


1-84 


940949 


4-25 


059051 


53 


8 


8i8io3 


2-41 


876899 

876789 


1-84 


9 4-2c4 


4-25 


058796 


5a 


9 


818247 


2-41 


1-84 


94 458 


4-25 


058542 


5i 


10 


8i83 9 2 


241 


876678 


1.84 


94^14 


4-25 


058286 


5o 


ii 


o-8i8536 


2-40 


9 -8 7 6568 


1-84 


9-041968 


4-25 


iO'058o3a 


8 


]2 


818681 


2-40 


876457 


1-84 


94*2*3 


4-25 


057777 


i3 


818825 


2-40 


8 7 6347 


1.84 


942478 


4-25 


057522 


47 


i4 


818969 
819113 


2-40 


876236 


i-85 


94-7 3 3 


4-25 


057267 


46 


l5 


2-40 


876125 


i-85 


9/ 988 


4-25 


057012 


45 


16 


819207 


2-40 


876014 


i-85 


94-243 


4-25 


056757 


44 


i7 


819401 


2 -40 


875904 


i-85 


943498 
943752 


4-25 


o565o2 


43 


18 


819545 
819689 


2.39 


8 7 5 79 3 


i-85 


4-25 


056248 


42 


19 


2-39 


875682 


i-85 


044007 


4-25 


055993 


4i 


20 


819832 


2-39 


8 7 55 7 i 


i-85 


944262 


4-25 


o55739 


40 


21 


9-819976 


2 -3 9 


9.875459 
.875348 


i-85 


9-944^17 


4-25 


io-o55483 


3 9 


22 


820120 


2.39 


i-85 


94^771 


4-24 


055229 


38 


23 


820263 


2-39 


S75237 


i-85 


945026 


4-24 


. 054974 


u 


24 


820406 


2-39 


875126 


i-86 


9^281 


4-24 


054719 
054465 


25 


82o55o 


2-38 


875014 


1.86 


945535 


4-24 


35 


26 


820693 
820836 


2-38 


874903 


1-86 


945790 


4-24 


054210 


34 


27 


2-33 


87479 1 


i-86 


946045 


4-24 


053955 


33 


28 


820979 


2-38 


874680 


i-86 


946299 


4-24 


053701 


32 


29 


821122 


2-33 


874568 


i-86 


946554 


4-24 


o53446 


3i 


3o 


821265 


2-38 


8 7 4456 


i-86 


946808 


4-24 


053192 


3o 


3i 


9-821407 


2-38 


9-874344 


1-86 


9-947063 


4-24 


10-052937 


29 


32 


82i55o 


2-38 


874232 


1 -87 


9473i8 


4-24 


052682 


28 


33 


821693 


2-3 7 


874121 


1-87 


947572 


4-24 


052428 


27 


34 


82i835 


2-3 7 


874009 


1-87 


947826 


4-24 


052174 


26 


35 


821977 


2-3 7 


8 7 38 9 6 


1-87 


948081 


4-24 


051919 


25 


36 


822120 


2-3 7 


873784 


1-87 


948336 


4-24 


051664 


24 


37 


822262 


2-3 7 


873672 


1.87 


948590 


4-24 


o5i4io 


23 


38 


822404 


2-3 7 


873560 


1-87 


948844 


4-24' 


o5n56 


22> 


3 9 


822546 


2,3 7 


873448 


1-87 


949099 


4-24 


050901 


21 


4o 


822688 


2-36 


873335 


1.87 


949353 


4-24 


050647 


20 


41 


9-822830 


2-36 


9-873223 


1.87 


9-949607 


4-24 


io-o5o393 


» 


42 


822972 


2-36 


873110 


1.88 


949862 


4-24 


o5oi38 


43 


823114 


2-36 


872998 


i-88 


9501 16 


4-24 


049884 


17 


44 


823255 


2-36 


872885 


i-88 


950370 


4-24 


049630 


16 


45 


823397 


2-36 


872772 


i-88 


950625 


4-24 


049375 


i5 


46 


823539 


2-36 


872659 


i-88 


950879 
95i i33 


4-24 


0491 2 1 


14 


47 


823680 


2-35 


872547 


i-88 


4-24 


048867 


i3 


48 


823821 


2-35 


872434 


i-88 


95i388 


4-24 


048612 


12 


49 


823 9 63 


2-35 


8 7 232i 


i-83 


951642 


4-24 


048358 


ii 


5o 


824104 


2-35 


872208 


i-88 


9 5i8 9 6 


4-24 


048104 


10 


5i 


9-824245 


2-35 


9-872095 


i-8 9 


9-952i5o 


4-24 


10 -047850 


8 


5j 


824386 


2-35 


871981 


1-89 


952405 


4-24 


047595 


53 


824527 


2-35 


871868 


1-89 


902659 


4 24 


047341 


7 


54 


824668 


2-34 


871755 


1-89 


952913 


4-24 


047087 


6 


55 


824808 


2-34 


871641 


1-89 


953167 


4-23 


046833 


5 


56 


824949 


2-34 


871528 


i-8g 


953421 


4-23 


046579 
046325 


4 


s 


820090 


2-34 


871414 


.•89 


953675 


4-23 


3 


53 


82523o 


2-34 


871301 


1.89 


953929 
9D4i83 


4-23 


04607 1 


2 


5 9 


825371 


2-34 


871187 


1-89 


4-23 


045817 


1 


60 


8a55n 


2-34 


871073 


1-90 


954437 


4-23 


045563 





Cosine 


~lJ. 


S-ine |48° 


Cotaug. 


P. 1 


Tang. _ 



L4 



60 



(42 DEGREES.) A TABLE OF LOGARITHMIC 



M. 




Sine 


D. 


Cosine | D. 


I Tang. 


1 D - 


| Cotang. 


— 


9-8255u 


2-34 


9-871073 


i-gc 


j 9-954437 


4-23 


10 -Q45563 


60 


i 

2 


82565i 
825791 


2-33 
2-33 


870960 
870846 ! 


1-90 

1-90 


934691 
954945 


4-23 
4-23 


o453oo 
o45o5t) 


u 


3 


825931 


2-33 


870732 


i-9< 


955200 


4-23 


044800 


57 


4 


826071 


2-33 


870618J 


i. 9 < 


955454 


4-23 


o44546 


56 


5 


82621 1 


2-33 


870504 


1-90 


955707 


4-23 


044293 


55 


6 


82635i 


2-33 


870390 


1-90 


955961 


4-23 


044039 

043785 


54 


7 


826491 


2-33 


-870276 


1-90 


9562i5 


4-23 


53 


8 


82663i 


2-33 


870161 


•9« 


956469 
956723 


4-23 


04353 1 


52 


9 


826770 


2-32 


870047 


■ 91 


4-23 


043277 


5i 


10 


826910 


2-32 


869933 


■ 91 


9 56 977 


4-23 


o43o23 


5o 


ii 


9.827049 


2-32 


9-869818 


■91 


9-957231 


4-23 


10-042769 
o425i5 


% 


12 


827189 


2-32 


869704 


•91 


957485 


4-23 


1 3 


827328 


2-32 


869589 


[•91 


9 5 77 39 
•957993 


4-23 


042261 


47 


14 


827467 


2-32 


869474 


[■91 


4-23 


042007 


46 


i5 


827606 


2-32 


869360 


1. 91 


958246 


4-23 


041754 


45 


16 


827745 


2-32 


869245 


[.91 


9585oo 


4-23 


o4i5oo 


44 


i? 


827884 


2-3l 


869 1 3o 


[•91 


958754 


4-23 


041246 


43 


18 


828023 


2-3l 


86901s 


.92 


959008 


4-23 


040992 


42 


»9 


8-8162 


2-3l 


868900 


.92 


959262 


4-23 


040738 


4i 


20 


8283oi 


2 -3 I 


868785 


.92 


959516 


4-23 


040484 


4o 


21 


9.828439 
828578 


2-3l 


9-868670 


[.92 


9-959769 


4-23 


io- o4o23 1 


u 


22 


2-3! 


868555 


[.92 


960023 


4-23 


3 9977 


23 


828716 


2-3l 


868440 


[•92 


960277 


4-23 


0397231 37 


24 


828855 


2-06 


868324 


[.92 


96o53i 


4-23 


039469 36 


25 


828993 
829131 


2-3o 


868209 
868093 


1-92 


960784 


4-23 


039216 
o38 9 62 


35 


26 


2-3o 


[.92 


9 6io38 


4-23 


34 


2 7 


829269 


2-3o 


867978 


i.93 


961291 


4-23 


038709 


33 


28 


829407 


2-3o 


867862 


.93 


961545 


4-23 


o3845o 


32 


29 


829545 


2-3o 


867747 


i- 9 3 


961799 
962052 


4-23 


o382oi 


3i 


3o 


8296S3 


2-3o 


86 7 63 1 


.. 9 3 


4-23 


o3 7 Q48 


3o 


3i 


9-829821 


2-29 


9-8675i5 


i. 9 3 


9-962306 


4-23 


10-037694 


3 


32 


829959 


2-29 


867399 
86 7 283 


- 9 3 


962560 


4-23 


037440 


33 


830097 
830234 


2-29 


[• 9 3 


962813 


4-23 


037187 


27 


34 


2-29 


867167 


- 9 3 


963067 


4-23 


036933 


26 


35 


83o372 


2-29 


867051 


- 9 3 


963320 


4-23 


o3668o 


25 


36 


83o5o9 


2-29 


866935 


.94 


9 635 7 4 


4-23 


o36426 


24 


u 


83o646 


2-29 


866819 
866703 


.94 


963827 


4-23 


036173 


23 


830784 


2-29 


• 94 


964081 


4-23 


035919 


22 


39 


830921 


2-28 


866586 


.94 


964335 


4-23 


o3566o 


21 


40 


83io58 


2-28 


^66470 


.94 


964588 


4-2$ 


o354i2 


20 


41 


9-83iig5 


2-28 


9-866353 


•94 


9-964842 


4-22 


io-o35i58 


3. 


42 


83i332 


2.28 


866237 


.94 


965095 


4-22 


03490 5 


43 


831469 
83 1 606 


2-28 


866120 


•94 


965349 


4-22 


o3465i 


]l 


44 


2-28 


866004 


• 9 5 


965602 


4-22 


034398 


45 


83i 7 42 


2-28 


865887 


- 9 5 


9 65855 


4-22 


o34i45 


i5 


46 


83 1 879 
832015 


2-28 


865770 


.95 


966105 


4-22 ' 


033891 
o33638 


14 


47 


2-27 


865653 i 


- 9 5 


9 66362 


4-22 


i3 


48 


832i5 2 


2-27 


865536 1 


. 9 5 


966616 


4-22 


o33384 


12 


49 


832288 


2-27 


865419 i 


- 9 5 


966869 


4-22 


o33i3i 


11 


5o 


832425 


2-27 


8653o2 1 


. 9 5 


967123 


4-22 


032877 


10 


5i 


9-83256i 


2-27 


9-865i85 1 


• 9 5 


9-967376 


4-22 


10-032624 


I 


52 


832697 


2-27 


865o68 1 


■ 9 5 


967629 


4-22 


032371 


53 


832833 


2-27 


864950 1 
864833 1 


• 95 


967883 
9 68i36 


4-22 


032117 


1 


54 


832969 
833 io5 


2-26 


.96 


4-22 


o3i864 


6 


55 


2-26 


864716 1 


.96 


968389 


4-22 


. o3i6u 


5 


56 


83324i 


2-26 


864598 1 
864481 1 


.96 


968643 


4-22 


o3i357 


4 


u 


8333 77 


2-26 


.96 


968896 


4-22 


o3no4! 3 


8335i2 


2-26 


864363 1 


• 96 


969149 
969403 


4-22 


o3o85i 


2 ; 


59 


833648 


2-26 


864245 1 


.96 


4-22 


o3o597 


I 


60 


833783 


a -26 


864127 1 


■06 


969656 


4-22 


o3o344 





I 


Cosine 


D. 


Pine 4 


tT' 


Cotansr. 


D. 1 


Tang. 


ML 



BINES AND TANGENTS. (43 DEGREES.) 



61 



to. 


Sine 


1 D> 


Cosine | D. 


Tan?. 


1 T) ' 


Cotane. 


r 





9-833783 


1 2-26 


9-864127JI.96 


9-969656 


4-22 


io-o3o344 


60 


I 


833919 


2-25 


8640 10 1-96 


96990c; 


4-22 


030091 1 59 


2 


834o54 


2-25 


863892 1 -97 


970162 


4-22 


o2 9 838| 58 


3 


834189 


2-25 


863774 1 -97 


970416 


4-22 


029584 


57 


4 


834323 


2-25 


863656i. 9 7 


970669 


4-22 


029331 


56 


5 


83446o 


1 2-25 


863538 1.97 


970922 


4-22 


02907? 


55 


6 


8345g5 
83473o 


2-25 


8634191 -97 


971175 


4-2? 


028825 


54 


7 


2-25 


8633ori-97 


97,1429 


4-22 


■ 028571 


53 


8 


834865 


2-25 


863i83.i-97 


971682 


4-22 


0283i8 


52 


9 


834999 


2-24 


863o64 ( i -97 


971935 


4-22 


028065 


5i 


10 


83 5 1 34 


2-24 


862946 1-98 


972188 


4-22 


.0278121 5o 


ii 


9.835269 


2-24 


9-862827 1.98 


9.972441 


4-2> 


10-027559! 49 


12 


8354o3 


2-24 


862709 1.98 


972694 


4-22 


0273061 48 


i3 


835538 


2-24 


862590 :i .98 


972948 


4-22 


027052! 47 


14 


8356 7 2 


2-24 


862471 i- 98 


973201 


4-22 


026799 46 


i5 


8358o 7 


2-24 


862353 1.98 


973454 


4-22 


026546 45 


16 


835 9 4i 


2-24 


86223411-98 


973707 


4-22 


0262931 44 


'7 


836o 7 5 


2-23 


862ii5 1 .98 


973960 


4-22 


026040 


43 


18 


8362oo 
836343 


2 : 23 


861996 i- 9 8 


9742i3 


4-22 


025787 


42 


19 


2-23 


861877 1.98 


974466 


4-22 


025534 


4i 


20 


836477 


2-23 


86175.8 1.99 


974719 


4-22 


025281 


40 


21 


9 .8366i 1 


2-23 


9-86i638;i- 9 9 


9-974973 


4-22 


10 025027 


3 9 


22 


836745 


2-23 


8615191.99 


975226 


4-22 


024774 


38 


23 


836878 


2-23 


861400! 1 -99 


97 5 479 


4-22 


024521 


37 


24 


837012 


2-22 


86I280 1 ! -99 


975732 


4-22 


024268 


36 


25 


807146 


2-22 


861161J1 -99 


975 9 85 


4-22 


024015 


35 


26 


837279 


S-22 


861041J1.99 


976238 


4-22 


023762 


34 


27 


83 14 1 2 


2-22 


86092211.99 
860802; 1 -99 


976491 


4-22 


023509I 33 


28 


837546 


2-22 


976744 


4-22 


023256! 32 


29 


83 7 6 79 


2-22 


860682 2-00 


976997 


4-22 


o23oo3i 3i 


3o 


83 7 8i2 


2-22 


86o562J2-oo 


977250 


4-22 


022750 3o 


3i 


9-837945 


2-22 


9-860442 2-00 


9-9775o3 


4-22 


10-022497 29 
022244 28 


'32 


838o 7 8 


2-21 


86o322 2-00 


977756 


4-22 


33 


83821 1 


2-21 


860202 2-00* 


978009 


4-22 


021991 


27 


34 


838344 


2-21 


860082 [2 -00 


978262 


4-22 


021738 


26 


35 


838477 


2-21 


8:^9962 2-00 


9 785i5 


4-22 


021485 


25 


36 


8386io 


2-21 


85984212-00 


978768 


4-22 


021232 


24 


37 


838742 


2-21 


859721 


2-01 


979021 


4-22 


020979 


23 


38 


838875 


2-21 


839601 


2-01 


979274 


4-22 


020726 22 


3 9 


839007 


2-21 


85 9 48o 


2-01 


979527 


4-22 


020473 1 21 


4o 


839140 


2-20 


85 Q 36o 


2-01 


979780 


4-22 


020220! 20 


4i 


9-839272 


2-20 


9*859239 


2-01 


9-98qo33 


4-22 


10-019967 IO 
019714! 10 


42 


839404 


2-20 


859II9 2-01 


980286 


4-22 


43 


83 9 536 


2-20 


858998,2-01 


9 8o538 


4-22 


OI9462! 17 , 


44 


83 9 668 


2- 20 


858877,2-01 


980791 


4-21 


OI9209! l6 


45 


83 9 8oo 


2-20 


858756 


2-02 


981044 


4-21 


018956 1 i5 


46 


839932 


2-20 


858635 


2-02 


981297 
9 8i55o 


4-21 


018703 14 
oi845o i3 


47 


840064 


2-I 9 


8585i4 


2-02 


4-21 


48 


840196 


2-19 


8583 9 3 


2-02 


981803 


4-21 


018197J 12 


4 9 


840328 


2.I9 


858272 


2-02 


9 8 2 o56 


4-21 


or7944 ; 11 


5o 


840459 


2-19 


858i5i 


2-02 


982309 


4-21 


017691 10 


5i 


9-840591 


2-lg 


9'858o2 9 
857908 


2-02 


9.982562 


4-21 


io-oi7438 ! 9 
01 7 1 86,' 8 


52 


840722 


2-19 


2-02 


982814 


4-21 


53 


840854 


2.I9 


85 77 86 


2-02 


983067 


4-21 


016933I 7 


54 1 


840985 


2-19 


857665 


2-03 


983320 


4-21 


0166801 6 


55 ! 


841 1 16 


2- 18 


85 7 543 1 2-o3 


9835 7 3 


4-2V 


016427 


5 


56 


841247 


2-lS 


857422 2-03| 


983826 


4-21 


016174 


4 


& 


84i3 7 8 


2- 1 3 


8573oo!2-o3l 


984079 


4-21 


015921 


3 


84:5o 9 


2-l8 


85 7 i 7 82- o3 


984331 


4-21 


015669 


2 


5 9 


841640 


2-l8 


857o56!2-o3 


984584 


4-21 


oi54i6 


1 


6o 


841771 


218 


856 9 34|2-o3 


984837 


4-21 


oi*i63 





Cosine 


D. 


S\uT~ liGo| 


Cotansr. 


D. 1 


Tang. 


M-, 



02 



SINES AND TANGENTS. 



M. 


Sine 


D. 


Cosine 


1). 


Tang. 


D. 


Cctang. 







"9^841771 


2-l8 


9.856934 


>-o3 


9-984837 


4-21 


io-oi5i63 


60 


i 


841902 


2-18 


8568i2 


!-oc 


985090 


4-21 


014910 


s- 


2 


842o33 


2-l8 


856690 


J- 04 


985343 


4-21 


014657 


3 


842i63 


2-17 


856568 ' 


1 -o4 


9 855 9 6 


4-21 


014404 


u 


4 


842294 


2-17 


856446 ' 


8- o4 


9 85848 


4-21 


014152 


5 


842424 


2-17 


856323 ' 


> - 04 


986101 


4-21 


013899 


55 


6 


842555 


2-17 


8562oi ' 


S-O^ 


986354 


4-21 


013646 


54 


I 


842685 


2-17 


856o 7 8 


I ■ 0/ 


986607 


4-21 


013393 


53 


8428i5 


2-17 


855 9 56 


I ■ 01 


986860 


4-21 


oi3i4o 


52 


9 


842946 


2-17 


855833 ' 


! • 01 


9871 j 2 


4-21 


012888 


5i 


IO 


843076 


2-17 


85571 1 


i-oi 


987365 


4-21 


012635 


5o 


II 


9-843206 


2-l6 


9-855588 


i-o5 


9.987618 


4-21 


IO-OI2382 


% 


12 


843336 


2-l6 


855465 


>-o5 


987871 


4-21 


012129 


i3 


843466 


2.16 


855342 ' 


>-n: 


988123 


4-21 


01 1 877 


47 


14 


843595 


2«l6 


855219 ' 


!-o5 


988376 


4-21 


01 1624 


46 


i5 


843725 


2.16 


855og6 : 


»-o5 


988629 


4-21 


011371 


45 


16 


843855 


2-l6 


854973 : 


•o5 


988882 


4-21 


011118 


44 


n 


843984 


2-l6 


85485o j 


s-o5 


989134 


4'2I 


010866 


43 


18 


844i 14 


2-l5 


854727 : 


>-o6 


989387 


4-21 


oio6i3 


42 


19 


844243 


2-l5 


8546o3 : 


>-o6 


989640 


4-21 


oio36o 


41 


20 


844372 


2-l5 


85448o : 


>-o6 


989893 


4-21 


010107 


40 


21 


9- 8445o2 


2-l5 


9-854356 : 


'•06 


9-990145 


4-21 


10-009855 


ii 


22 


84463 1 


2-l5 


854233 : 


• •06 


990398 
99063 1 


4-21 


009602 


23 


844760 


a.i5 


854ioo : 


• 06 


4-21 


009349 


iz 


24 


844889 
845oi8 


2-l5 


853o86 : 


>-o6 


990903 


4-21 


009097 


25 


2-l5 


853862 : 


► •06 


99 1 1 56 


4-21 


008844 


35 


26 


845 1 47 


2-l5 


853 7 38 : 


•o(: 


991409 


4-21 


008591 


34 


27 


843276 


2-14 


8536i4 : 


•07 


991662 


4-21 


oo8338 


33 


28 


8454o5 


2-14 


853490 2 


•07 


991914 


4-21 


008086 


32 


29 


845533 


2-H 


853366 : 


•07 


992167 


4-21 


007833 


3i 


3o 


845662 


2-14 


853242 : 


>-07 


992420 


4-21 


007580 


3o 


3i 


9-845790 


2-!4 


9 .853iiS ; 


•07 


9-992672 


4-21 


10-007328 


8 


32 


845919 


2-14 


852994: 
85286gh; 

852745k 


•07 


992925 


4-21 


007075 


33 


846047 


2-14 


•07 


993178 


4-21 


006822 


27 


34 


846175 


2-14 


• 07 


093430 


4-21 


006570 


26 


35 


846364 


2-14 


85 2620 2 


•07 


993683 


4-21 


006317 


25 


36 


846432 


2-l3 


852496 5 


.08 


993936 


'4-21 


006064 


24 


?7 


84656o 


2-l3 


852371 2 


.08 


994189 


4-21 


oo58u 


23 


38 


846688 


2-l3 


852247 s 


.08 


994441 


4-21 


oo5559 


22 


3 9 


846816 


2-l3 


852122 2 


•08 


994694 


4-21 


oo53o6 


21 


4o 


846944 


2-l3 


85i997 2 


•08 


994947 


4-21 


oo5o53 


20 


4i 


9-847071 


2-l3 


9-851872 2 


.08 


9-995199 
995452 


4-21 


10-004801 


;g 


42 


847199 


2-l3 


851747 2 


•08 


4-21 


004548 


43 


847327 


2-l3 


85l622 2 


•08 


995705 


4-21 


004295 


ii 


44 


847454 


2-12 


85 1 497 2 


•09 


9 9 5 9 57 


4-21 


004043 


45 


847582 


2-12 


8513722 


•09 


996210 


4-21 


003790 
oo3537 


i5 


46 


847709 


2-12 


8512462 


.09 


996463 


4-21 


14 


47 


847836 


2-12 


85ii2i 2 


•09 


996715 


4-21 


oo3285 


i3 


48 


847964 


2-12 


85099612 
85o870 ! 2 


•09 


996968 


4-21 


oo3o32 


12 


^ 9 


848091 


2-12 


•09 


997221 


4-21 


002779 


11 


5o 


848218 


2-12 


85o745J2 


•09 


997473 


4-21 


002527 


10 


5i 


9-848345 


2-12 


9«85o6i9 2 


•09 


9.997726 


4-2! 


10-002274 


I 


52 


848472 


2-II 


85o4g3 2 


• FO 


997979 


4-21 


00202 i 


53 


848099 


2-II 


85o368 2 


•10 


098231 


4-21 


001769 


I 


54 


848726 


2-II 


85o242|2 


• ro 


998484 


4-21 


ooi5i6 


55 


848852 


2-II 


85on6a 


• 10 


998737 


4-21 


001263 


5 


56 


848979 


2-II 


8499902 
84986412 


• TO 


998989 


4-21 


OOIOII 


4 


n 


849106 


2-II 


• IO 


999242 


4-21 


000758 


3 


849232 


2-H 


8497382 


• ro 


999495 


4-21 


ooo5o5 


a 


59 


849359 
849485 


2 • I I 


849611(2 


• 10 


999748 


4-21 


000 2 53 


1 


Co 


2-H 


849485 2 


• 10 


10- 000000 


4-21 


10- 000000 







Cosine 


D. 


Sine |4 5° 


Cotang. 


I). 


Tang. 


JL 



